Complexity and Infinite Games on Finite Graphs

Complexity and Infinite Games on Finite Graphs
Complexity and Infinite Games
on Finite Graphs
Paul William Hunter
University of Cambridge
Computer Laboratory
Hughes Hall
July 2007
This dissertation is submitted for the degree of Doctor of Philosophy
Declaration
This dissertation is the result of my own work and includes nothing which is the
outcome of work done in collaboration except where specifically indicated in the text.
This dissertation does not exceed the regulation length of 60 000 words, including
tables and footnotes.
Complexity and Infinite Games on Finite Graphs
Paul William Hunter
Summary
This dissertation investigates the interplay between complexity, infinite games, and
finite graphs. We present a general framework for considering two-player games on
finite graphs which may have an infinite number of moves and we consider the computational complexity of important related problems. Such games are becoming increasingly important in the field of theoretical computer science, particularly as a tool
for formal verification of non-terminating systems. The framework introduced enables
us to simultaneously consider problems on many types of games easily, and this is
demonstrated by establishing previously unknown complexity bounds on several types
of games.
We also present a general framework which uses infinite games to define notions
of structural complexity for directed graphs. Many important graph parameters, from
both a graph theoretic and algorithmic perspective, can be defined in this system. By
considering natural generalizations of these games to directed graphs, we obtain a novel
feature of digraph complexity: directed connectivity. We show that directed connectivity is an algorithmically important measure of complexity by showing that when it is
limited, many intractable problems can be efficiently solved. Whether it is structurally
an important measure is yet to be seen, however this dissertation makes a preliminary
investigation in this direction.
We conclude that infinite games on finite graphs play an important role in the area
of complexity in theoretical computer science.
Acknowledgements
A body of work this large can rarely be completed without the assistance and support of many others. Any effort to try and acknowledge all of them would undoubtedly
result in one or two being left out, so I am only able to thank those that feature most
prominently in my mind at the moment.
The three people I am most indebted to fit nicely into the categories of my past, my
present, and my future. Starting with my future (it always pays to look forwards), I am
particularly grateful to Stephan Kreutzer. Working with you for a year in Berlin was
a fantastic experience and I am looking forward to spending the next few years in the
same city again. Thank you for all your support and guidance.
To Sarah, with your constant encouragement (some would say nagging), I am forever indebted. Without your care and support I might never have finished. And I would
certainly not be where or who I am today without you.
Finally, the submission of this thesis ends my formal association with the person
to whom I, and this dissertation, owe the most gratitude: my supervisor Anuj Dawar.
Thank you for giving me the opportunity to work with you and for setting me on the
path to my future. You have been an inspiration as a supervisor, I can only hope that
when it is my turn to supervise PhD students I can live up to the example you have set
me.
Contents
1
Introduction
Notation and Conventions . . . .
1.1.1 Sets and sequences
1.1.2 Graphs . . . . . .
1.1.3 Complexity . . . .
Collaborations . . . . . . . . . .
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1
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2 Infinite games
2.1 Preliminaries . . . . . . . . . . . . . . . .
2.1.1 Arenas . . . . . . . . . . . . . . .
2.1.2 Games . . . . . . . . . . . . . . .
2.1.3 Strategies . . . . . . . . . . . . . .
2.1.4 Simulations . . . . . . . . . . . . .
2.2 Winning condition presentations . . . . . .
2.2.1 Examples . . . . . . . . . . . . . .
2.2.2 Translations . . . . . . . . . . . . .
2.2.3 Extendibility . . . . . . . . . . . .
2.3 Complexity results . . . . . . . . . . . . .
2.3.1 P SPACE-completeness . . . . . . .
2.3.2 Complexity of union-closed games
2.4 Infinite tree automata . . . . . . . . . . . .
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11
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3 Strategy Improvement for Parity Games
3.1 The strategy improvement algorithm . . . . . . . . . . . . . . . . . .
3.2 A combinatorial perspective . . . . . . . . . . . . . . . . . . . . . .
3.3 Improving the known complexity bounds . . . . . . . . . . . . . . .
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4 Complexity measures for digraphs
4.1 Tree-width . . . . . . . . . . . . . . . . . . . .
4.1.1 Structural importance of tree-width . .
4.1.2 Algorithmic importance of tree-width .
4.1.3 Extending tree-width to other structures
4.2 Directed tree-width . . . . . . . . . . . . . . .
4.3 Beyond directed tree-width . . . . . . . . . . .
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CONTENTS
x
5 Graph searching games
5.1 Definitions . . . . . . . . . . . . . . . . . .
5.1.1 Strategies . . . . . . . . . . . . . .
5.1.2 Simulations . . . . . . . . . . . . .
5.2 Examples . . . . . . . . . . . . . . . . . .
5.2.1 Cops and visible robber . . . . . .
5.2.2 Cops and invisible robber . . . . .
5.2.3 Cave searching . . . . . . . . . . .
5.2.4 Detectives and robber . . . . . . . .
5.2.5 Cops and inert robber . . . . . . . .
5.2.6 Cops and robber games . . . . . . .
5.3 Complexity measures . . . . . . . . . . . .
5.3.1 Example: Cops and visible robber .
5.3.2 Example: Cops and invisible robber
5.3.3 Example: Cops and inert robber . .
5.3.4 Example: Other resource measures
5.3.5 Monotonicity . . . . . . . . . . . .
5.4 Robustness results . . . . . . . . . . . . . .
5.4.1 Subgraphs . . . . . . . . . . . . . .
5.4.2 Connected components . . . . . . .
5.4.3 Lexicographic product . . . . . . .
5.5 Complexity results . . . . . . . . . . . . .
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69
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6 DAG-width
6.1 Cops and visible robber game . . . . . . . . . . . . . . .
6.1.1 Monotonicity . . . . . . . . . . . . . . . . . . .
6.2 DAG-decompositions and DAG-width . . . . . . . . . .
6.3 Algorithmic aspects of DAG-width . . . . . . . . . . . .
6.3.1 Computing DAG-width and decompositions . . .
6.3.2 Algorithms on graphs of bounded DAG-width . .
6.3.3 Parity Games on Graphs of Bounded DAG-Width
6.4 Relation to other graph connectivity measures . . . . . .
6.4.1 Undirected tree-width . . . . . . . . . . . . . .
6.4.2 Directed tree-width . . . . . . . . . . . . . . . .
6.4.3 Directed path-width . . . . . . . . . . . . . . .
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7 Kelly-width
7.1 Games, orderings and k-DAGs . . . . . . . . . . .
7.1.1 Inert robber game . . . . . . . . . . . . . .
7.1.2 Elimination orderings . . . . . . . . . . .
7.1.3 Partial k-trees and partial k-DAGs . . . . .
7.1.4 Equivalence results . . . . . . . . . . . . .
7.2 Kelly-decompositions and Kelly-width . . . . . . .
7.3 Algorithmic aspects of Kelly-width . . . . . . . . .
7.3.1 Computing Kelly-decompositions . . . . .
7.3.2 Algorithms on graphs of small Kelly-width
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CONTENTS
7.4
xi
7.3.3 Asymmetric matrix factorization . . . . . . . . . . . . . . . .
Comparing Kelly-width and DAG-width . . . . . . . . . . . . . . . .
8 Havens, Brambles and Minors
8.1 Havens and brambles . . . . . . . . . . . .
8.2 Directed minors . . . . . . . . . . . . . . .
8.2.1 What makes a good minor relation?
8.2.2 Directed minor relations . . . . . .
8.2.3 Preservation results . . . . . . . . .
8.2.4 Algorithmic results . . . . . . . . .
8.2.5 Well-quasi order results . . . . . .
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9 Conclusion and Future work
9.1 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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References
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xii
CONTENTS
Chapter 1
Introduction
The aim of this dissertation is to investigate the interplay between infinite games, finite
graphs, and complexity. In particular, we focus on two facets: the computational complexity of infinite games on finite graphs, and the use of infinite games to define the
structural complexity of finite graphs. To present the motivation behind this investigation, we consider the three fundamental concepts of games, graphs and complexity.
What is a game?
Ask anyone what a game is and most people will respond with an example: chess,
bridge, cricket, and so on. Almost everyone understands what a game is, but few
people can immediately give a precise definition. Loosely speaking, a game involves
interactions between a number of players (possibly only one) with some possible outcomes, though the outcome is not always the primary concern. The importance of
games in many scientific fields arises from their usefulness as an informal description
of systems with complex interactions; as most people understand games, a description
in terms of a game can often provide a good intuition of the system. The prevalence of
this application motivates the formal study of games, which results in the use of games
to provide formal definitions. Such definitions can sometimes provide interpretations
of concepts where traditional approaches are cumbersome or less than adequate. For
example, the semantics of Hintikka’s Independence Friendly logic [HS96] are readily
expressed using games of imperfect information, but the traditional Tarski-style approaches are unwieldy.
Games in computer science
Mathematical games are playing an increasingly important role in computer science,
both as informal descriptions and formal definitions. For example, tree-width, an algorithmically important graph parameter which we see frequently in this dissertation,
can be intuitively presented as a game in which a number of cops attempt to capture
a robber on a graph. Examples where games can provide formal definitions include
interactive protocols and game semantics. An important example of an application of
1
2
CHAPTER 1. INTRODUCTION
games, which motivates the games we consider, is the following game that arises when
verifying if a system satisfies certain requirements.
Starting with the simple case of checking if a formula of propositional logic is satisfied by a truth assignment, consider the following game played by two players, Verifier
and Falsifier, “on” the formula. The players recursively choose subformulas with Verifier choosing disjuncts and Falsifier choosing conjuncts until a literal is reached. If
the truth value of that literal is true then Verifier wins, otherwise Falsifier wins. The
formula is satisfiable if, and only if, Verifier has a strategy to always win. This game
is easily extended to the verification of first order formulas, with Verifier choosing elements bounded by existential quantifiers and Falsifier choosing elements bounded by
universal quantifiers. Verifying a first order logic formula is very useful for checking
properties of a static system, but often in computer science we are also interested in
formally verifying properties of reactive systems, systems which interact with the environment and change over time. Requirements for such systems are often specified
in richer logics such as Linear Time Logic (LTL), Computation Tree Logic (CTL) or
the modal µ-calculus. This motivates the following extension of the Verifier-Falsifier
game for verifying if a reactive system satisfies a given set of requirements. The game
is played by two players, System and Environment, on the state space of the reactive
system. The current state of the system changes, either as a consequence of some move
effected by Environment, or some response by System. System takes the role of Verifier, trying to keep the system in a state which satisfies the requirements to be verified.
Environment endeavours to demonstrate the system does not satisfy the requirements
by trying to move the system into a state which does not satisfy the requirements.
The natural abstraction of these games is a game where two players move a token
around a finite directed graph for a possibly infinite number of moves with the winner
determined by some pre-defined condition. This abstraction encompasses many twoplayer, turn-based, zero-sum games of perfect information, and such games are found
throughout computer science: in addition to the games associated with formal verification of reactive systems, examples of games which can be specified in this manner
include Ehrenfeucht-Fraı̈ssé games and the cops and robber game which characterizes
tree-width. Unsurprisingly, these games have been extensively researched, particularly
in the area of formal verification: see for example [BL69, Mul63, EJ88, Mos91, EJ91,
IK02, DJW97]. Two important questions regarding the complexity of such games are
left unresolved in the literature. These are the exact complexity of deciding Muller
games and the exact complexity of deciding parity games. One of the goals of this
dissertation is to address these questions with an investigation of the computational
complexity of deciding the winner of these types of games.
What is a graph?
Graphs are some of the most important structures in discrete mathematics. Their ubiquity can be attributed to two observations. First, from a theoretical perspective, graphs
are mathematically elegant. Even though a graph is a simple structure, consisting only
of a set of vertices and a relation between pairs of vertices, graph theory is a rich and
varied subject. This is partly due the fact that, in addition to being relational structures,
graphs can also be seen as topological spaces, combinatorial objects, and many other
3
mathematical structures. This leads to the second observation regarding the importance of graphs: many concepts can be abstractly represented by graphs, making them
very useful from a practical viewpoint. From an algorithmic point of view, many problems can be abstracted to problems on graphs, making the study of graph algorithms a
particularly fruitful line of research.
In computer science, many structures are more readily represented by directed
graphs, for example: transition systems, communications networks, or the formal verification game we saw above. This means that the study of directed graphs and algorithms for directed graphs is particularly important to computer science. However, the
increased descriptive power of directed graphs comes at a cost: the loss of symmetry
makes the mathematical theory more intricate. In this dissertation we explore both the
algorithmic and mathematical aspects of directed graphs.
What is complexity?
Just as the definition of a game is difficult to pin down, the quality of “being complex” is
best described by examples and synonyms. From an algorithmic perspective, a problem
is more complex than another problem if the latter is easier to compute than the former.
From a structural point of view, one structure is more complex than another if the first
structure contains more intricacies. These are the two kinds of complexity relevant to
this dissertation: computational complexity and structural complexity.
In the theory of algorithms, the notion of computational complexity is well defined.
In model theory however, being structurally complex is very much a subjective notion,
depending largely on the application one has in mind. For example, a graph with a large
number of edges could be considered more complex than a graph with fewer edges. On
the other hand, a graph with a small automorphism group could be considered more
complex than a graph with a large automorphism group, as the second graph (which
may well have more edges) contains a lot of repetition. As we are primarily interested
in algorithmic applications in this dissertation, we focus on the structural aspects of
graphs which influence the difficulty of solving problems. In Section 1.1.2 below, we
loosely define this notion of graph structure by describing the fundamental concepts
important in such a theory.
Having established what constitutes “structure”, we turn to the problem of defining
structural complexity. The most natural way is to define some sort of measure which
gives an intuition for how “complex” a structure is. In Chapter 4, we discuss those
properties that a good measure of structural complexity should have. But how do we
find such measures in the first place? Also in Chapter 4 we present the notion of treewidth and argue that it is a good measure of complexity for undirected graphs. As
we remarked above, tree-width has a characterization in terms of a two-player game,
so it seems that investigating similar games would yield useful measures for structural
complexity. Indeed this has been an active area of research for the past few years, for
example: [KP86, LaP93, ST93, DKT97, JRST01, FT03, FFN05, BDHK06, HK07].
This line of research has recently started to trend away from showing game-theoretic
characterizations of established structural complexity measures to defining important
parameters from the definition of the game, an example of the transition from the use
of games as an informal description to their use as a formal definition. Despite this
4
CHAPTER 1. INTRODUCTION
activity, very little research has considered games on directed graphs. This is perhaps
partly due to the lack, for some time, of a reasonable measure of structural complexity
for directed graphs.
The second major goal of this dissertation is to use infinite games to define a notion of
structural complexity for directed graphs which is algorithmically useful.
Organization of the thesis
In the remainder of this chapter we define the conventions we use throughout. Chapters 2 and 3 are primarily concerned with the analysis of the complexity of deciding the
winner of infinite games on finite graphs. From Chapter 4 to Chapter 8 we investigate
graph complexity measures defined by infinite games.
In Chapter 2 we formally define the games we are interested in. We introduce the
notion of a winning condition type and we establish a framework in which the expressiveness and succinctness of different types of winning conditions can be compared.
We show that the problem of deciding the winner in Muller games is P SPACE-complete,
and use this to show the non-emptiness and model-checking problems for Muller tree
automata are also P SPACE-complete.
In Chapter 3 we analyse an algorithm for deciding parity games, the strategy improvement algorithm of [VJ00a]. We present the algorithm from a combinatorial perspective, showing how it relates to finding a global minimum on an acyclic unique sink
oriented hypercube. We combine this with results from combinatorics to improve the
bounds on the running time of the algorithm.
In Chapter 4 we discuss the problem of finding a reasonable notion of complexity
for directed graphs. We present the definition of tree-width, arguably one of the most
practical measures of complexity for undirected graphs, and we discuss the problem of
extending the concept to directed graphs.
Building on the games defined in Chapter 2, in Chapter 5 we define the graph
searching game. We show how we can use graph searching games to define robust
measures of complexity for both undirected and directed graphs. This framework is
general enough to include many examples from the literature, including tree-width.
In Chapters 6 and 7 we introduce two new measures of complexity for directed
graphs: DAG-width and Kelly-width. Both arise from the work in Chapter 5, and both
are generalizations of tree-width to directed graphs. While DAG-width is arguably
the more natural generalization of the definition of tree-width, Kelly-width is equivalent to natural generalizations of other graph parameters equivalent to tree-width on
undirected graphs, which we also introduce in Chapter 7. We show each measure is
useful algorithmically by providing an algorithm for deciding parity games which runs
in polynomial time on the class of directed graphs of bounded complexity. We compare both measures with other parameters defined in the literature such as tree-width,
directed tree-width and directed path-width and show that these measures are markedly
different to those already defined. Finally, in Chapter 7 we compare Kelly-width and
DAG-width. We show that the two measures are closely related, but we also show that
there are graphs on which the two measures differ.
In Chapter 8 we present some preliminary work towards a graph structure theory
for directed graphs based on DAG-width and Kelly-width. We define generalizations
5
of havens and brambles which seem to be appropriate structural features present in
graphs of high complexity and absent in graphs of low complexity. We also consider
the problem of generalizing the minor relation to directed graphs.
We conclude the dissertation in Chapter 9 by summarizing the results presented.
We discuss the contribution made towards the stated research goals, and consider directions of future research arising from this body of work.
Notation and Conventions
We assume the reader is familiar with basic complexity theory, graph theory and discrete mathematics. We generally adopt the following conventions for naming objects.
• For elementary objects, or objects we wish to consider elementary, for example
vertices or variables: a, b, c, . . .
• For sets of elementary objects: A, B, C, . . .
• For structures comprising several sets, including graphs and families of sets:
A, B, C, . . .
• For more complex structures: A, B, C, . . .
• For sequences and simple functions: α, β, γ, . . .
• For more complex functions: A, B, C, . . .
1.1.1 Sets and sequences
All sets and sequences we consider in this dissertation are countable. We use both N
and ω to denote the natural numbers, using the latter when we require the linear order.
We also assume that 0 is a natural number.
Let A be a set. We denote by P(A) the set of subsets of A. For a natural number k,
[A]k denotes the set of subsets of A of size k, and [A]≤k denotes the set of subsets of
˙ denotes their disjoint union and A 4 B
A of size ≤ k. Given two sets A and B, A∪B
denotes their symmetric difference. That is,
A 4 B := (A \ B) ∪ (B \ A).
For readability, we generally drop innermost parentheses or brackets when the intention
is clear, particularly with functions. For example if f : P(A) → B, and a ∈ A, we
write f (a) for f ({a}).
We write sequences as words a1 a2 · · · , using 0 as the first index when the first
element of the sequence is especially significant. For a sequence π, |π| denotes the
length of π (|π| = ω if π is infinite). We denote sequence concatenation by ·. That
is, if π = a1 a2 · · · an is a finite sequence and π 0 = b1 b2 · · · is a (possibly infinite)
sequence, then π · π 0 is the sequence a1 a2 · · · an b1 b2 · · · . If π = a1 a2 · · · an is a finite
sequence, π ω is the infinite sequence π · π · π · · · . Given a set A, the set A∗ denotes the
set of all finite sequences of elements of A, and the set Aω denotes the set of all infinite
CHAPTER 1. INTRODUCTION
6
sequences. We say a reflexive and transitive relation ≤ on A is a well-quasi ordering if
for any infinite sequence a1 a2 · · · ∈ Aω , there exists indices i < j such that xi ≤ xj .
Let π = a1 a2 · · · and π 0 = b1 b2 · · · be sequences of elements of A. We write
π π 0 if π is a prefix of π 0 , that is, if there exists a sequence π 00 such that π 0 = π · π 00 .
We write π ≤ π 0 if π is a subsequence of π 0 , that is, there exists a sequence of natural
numbers n1 < n2 < · · · such that ai = bni for all i ≤ |π|.
1.1.2 Graphs
The notation we use for the graph theoretical aspects of this dissertation generally
follow Diestel [Die05], however rather than regarding directed graphs as undirected
graphs with two maps Head and Tail from edges to vertices, we view directed graphs
as relational structures. That is, a directed graph, or digraph, G consists of a set of
vertices, denoted V (G), and an edge relation, E(G) ⊆ V (G) × V (G). We use the
definition in [Die05] for an undirected graph, that is E(G) is a subset of [V (G)]2 . For
an edge e = (u, v) in a directed graph, the head of e is v and the tail is u, and we say
e goes from u to v. To avoid ambiguities, we assume that the vertex and edge sets are
disjoint. The elements of a graph G, is the set defined as
Elts(G) := V (G) ∪ E(G).
We note that we could either adopt the policy of Diestel and view a directed graph
as an undirected graph with some additional structural information, or alternatively we
could view an undirected graph as a directed graph where the edge relation is symmetric and irreflexive. We reserve those interpretations for the following two maps
between directed and undirected graphs. Let D be a directed graph. The underlying
undirected graph of D is the undirected graph D where:
• V (D) = V (D), and
• E(D) = {u, v} : (u, v) ∈ E(D)}.
←
→
Let G be an undirected graph. The bidirected graph of G is the directed graph G
where:
←
→
• V ( G ) = V (G), and
←
→
• E( G ) = (u, v), (v, u) : {u, v} ∈ E(G)}.
We extend the definition of bidirection to parts of undirected graphs. For example a
bidirected cycle is a subgraph of a directed graph which is a bidirected graph of a cycle.
Regarding the pair of edges {(u, v), (v, u)} arising from bidirecting an undirected edge,
we call such a pair anti-parallel. For clarity when illustrating directed graphs, we use
undirected edges to represent pairs of anti-parallel edges. For the remaining definitions,
we use ordered pairs to describe edges in undirected graphs.
Let G be an undirected (directed) graph. A (directed) path in G is a sequence of
vertices π = v1 v2 · · · such that for all i, 1 ≤ i < |π|, (vi , vi+1 ) ∈ E(G). For a subset
7
X ⊆ V (G), the set of vertices reachable from X is defined as:
ReachG (X) := {w ∈ V (G) : there is a (directed) path to w from some v ∈ X}.
For a subset X ⊆ V (G) of the vertices, the subgraph of G induced by X is the
undirected (directed) graph G[X] defined as:
• V (G[X]) = X, and
• E(G[X]) = {(u, v) ∈ E(G) : u, v ∈ X}.
For convenience we write G \ X for the induced subgraph G[V (G) \ X]. Similarly, for
a set E of edges, G[E] is the subgraph of G with vertex set equal to the set of endpoints
of E, and edge set equal to E.
Let v ∈ V (D) be a vertex of a directed graph D. The successors of v are the
vertices w such that (v, w) ∈ E(D). The predecessors of v are the vertices u such
that (u, v) ∈ E(D). The successors and predecessors of v are the vertices adjacent
to v. We say v is a root (of D) if it has no predecessors, and a leaf (of D) if it has
no successors. The outgoing edges of v are all the edges from v to some successor of
v, and the incoming edges of v are all the edges from a predecessor of v to v. The
outdegree of v, dout (v) is the number of outgoing edges of v and the indegree of v,
din (v) is the number of incoming edges of v. Given a subset V ⊆ V (G) of vertices, the
out-neighbourhood of V , Nout (V ) is the set of successors of vertices of V not contained
in V .
If D is a directed acyclic graph (DAG), we write D for the reflexive, transitive
closure of the edge relation. That is v D w if, and only if, w ∈ ReachD (v). If
v D w, we say v is a ancestor of w and w is a descendant of v.
We denote by Dop the directed graph obtained by reversing the directions of the
edges of D. That is, Dop is the directed graph defined as:
• V (Dop ) = V (D), and
• E(Dop ) = (E(D))−1 = {(v, u) : (u, v) ∈ E(D)}.
In this dissertation we consider transition systems with a number of transition relations. That is, a transition system is a tuple (S, sI , E1 , E2 . . .) where S is the set of
states, sI ∈ S is the initial state, and Ei ⊆ S × S are the transition relations. We
observe that a transition system with one transition relation is equivalent to a directed
graph with an identified vertex.
Structural relations
As we indicated earlier, the notion of graph structure is very much a qualitative concept. Just as the “structure” of universal algebra is best characterized by subalgebras,
homomorphisms and products, the particular graph structure theory we are interested
in is perhaps best characterized by the following “fundamental” relations: subgraphs,
connected components and graph composition. As these concepts are frequently referenced, we include their definitions. First we have the subgraph relation.
CHAPTER 1. INTRODUCTION
8
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Figure 1.1: The lexicographic product of graphs G and H
Definition 1.1 (Subgraph). Let G and G 0 be directed (undirected) graphs. We say G is
a subgraph of G 0 if V (G) ⊆ V (G 0 ) and E(G) ⊆ E(G 0 ).
The next definition describes the building blocks of a graph, the connected components.
Definition 1.2 (Connected components). Let G be an undirected graph. We say G is
connected if for all v, w ∈ V (G), w ∈ ReachG (v). A connected component of G is a
maximal connected subgraph.
It is easy to see that an undirected graph is the union of its connected components.
Sm
That is, if G1 ,S. . . , Gm are the connected components of G, then V (G) = i=1 V (Gi )
m
and E(G) = i=1 E(Gi ). From the maximality of a connected component, it follows
that a connected component is an induced subgraph. Thus we often view a connected
component as a set of vertices rather than a graph.
The final fundamental relation is lexicographic product, also known as graph composition.
Definition 1.3 (Lexicographic product). Let G and H be directed (undirected) graphs.
The lexicographic product of G and H is the directed (undirected) graph, G •H, defined
as follows:
• V (G • H) = V (G) × V (H), and
• (v, w), (v 0 , w0 ) ∈ E(G • H) if, and only if, (v, v 0 ) ∈ E(G) or v = v 0 and
(w, w0 ) ∈ E(H).
Intuitively, the graph G • H arises from replacing vertices in G with copies of H,
hence the name graph composition. Figure 1.1 illustrates an example of the lexicographic product of two graphs.
For directed graphs we have three more basic structural concepts: weakly connected components, strongly connected components and directed union. The first two
are a refinement of connected components.
Definition 1.4 (Weakly/Strongly connected components). Let G be a directed graph.
We say G is weakly connected if G is connected. We say G is strongly connected if for
all v, w ∈ V (G), w ∈ ReachG (v) and v ∈ ReachG (w). A weakly (strongly) connected
component of G is a maximal weakly (strongly) connected subgraph.
9
We observe that a directed graph is the union of its weakly connected components.
The union of the strongly connected components may not include all the edges of the
graph. However, it is easy to see that if there is an edge from one strongly connected
component to another, then there are no edges in the reverse direction. This leads to
the third structural relation specific to directed graphs.
Definition 1.5 (Directed union). Let G, G1 , and G2 be directed graphs. We say G is a
directed union of G1 and G2 if:
• V (G) = V (G1 ) ∪ V (G2 ), and
• E(G) ⊆ E(G1 ) ∪ E(G2 ) ∪ (V (G1 ) × V (G2 )).
It follows that a directed graph is a directed union of its strongly connected components.
1.1.3 Complexity
The computational complexity definitions of this dissertation follow [GJ79]. We consider polynomial time algorithms efficient, so we are primarily concerned with polynomial time reductions. We use standard big-O notation to describe asymptotically
bounded classes of functions, particularly for describing complexity bounds.
Collaborations
The work in several chapters of this dissertation arose through collaborative work with
others and we conclude this introduction by acknowledging these contributions. The
work regarding winning conditions in Chapter 2 was joint work with Anuj Dawar
and was presented at the 30th International Symposium on Mathematical Foundations
of Computer Science [HD05]. Chapter 6 arose through collaboration with Dietmar
Berwanger, Anuj Dawar and Stephan Kreutzer, and was presented at the 23rd International Symposium on Theoretical Aspects of Computer Science [BDHK06]. The concept and name DAG-width were also independently developed by Jan Obdržálek [Obd06].
Finally, the work in Chapter 7 arose through collaboration with Stephan Kreutzer and
was presented at the 18th ACM-SIAM Symposium on Discrete Algorithms [HK07].
10
CHAPTER 1. INTRODUCTION
Chapter 2
Infinite games
In this chapter we formally define the games we use throughout this dissertation. The
games we are interested in are played on finite or infinite graphs (whose vertices represent a state space) with two players moving a token along the edges of the graph. The
(possibly) infinite sequence of vertices that is visited constitutes a play of the game,
with the winner of a play being defined by some predetermined condition. As we discussed in the previous chapter, such games are becoming increasingly important in
computer science as a means for modelling reactive systems; providing essential tools
for the analysis, synthesis and verification of such systems.
It is known [Mar75] that under some fairly general assumptions, these games are
determined. That is, for any game one player has a winning strategy. Furthermore, under the conditions we consider below, the games we consider are decidable: whichever
player wins can be computed in finite time [BL69]. We are particularly interested in
the computational complexity of deciding which player wins in these games. Indeed,
this forms one of the underlying research themes of this dissertation.
As we are interested in the algorithmic aspects of these games, we need to restrict
our attention to games that can be described in a finite fashion. This does not mean that
the graph on which the game is played is necessarily finite as it is possible to finitely
describe an infinite graph. Nor does having a finite game graph by itself guarantee
that the game can be finitely described. Even with two nodes in a graph, the number
of distinct plays can be uncountable and there are more possible winning conditions
than one could possibly describe. Throughout this dissertation, we are concerned with
Muller games played on finite graphs. These are games in which the graph is finite
and the winner of a play is determined by the set of vertices of the graph that are
visited infinitely often in the play (see Section 2.1 for formal definitions). This category
of games is wide enough to include most kinds of game winning conditions that are
considered in the literature, including Streett, Rabin, Büchi and parity games.
Since the complexity of a problem is measured as a function of the length of the
description, the complexity of deciding which player wins a game depends on how
exactly the game is described. In general, a Muller game is defined by a directed
graph A, and a winning condition F ⊆ P(V (A)) consisting of a set of subsets of
V (A). One could specify F by listing all its elements explicitly (we call this an explicit
11
12
CHAPTER 2. INFINITE GAMES
presentation) but one could also adopt a formalism which allows one to specify F
more succinctly. In this chapter we investigate the role the specification of the winning
condition has in determining the complexity of deciding regular games. Examples of
this line of research can be found throughout the literature, for instance the complexity
of deciding Rabin games is known to be NP-complete [EJ88], for Streett games it is
known to be co-NP-complete. The complexity of deciding parity games is a central
open question in the theory of regular games. It is known to be in NP ∩ co-NP [EJ91]
and conjectured by some to be in P TIME. In Chapters 3, 6 and 7 we explore this
problem in more detail. For Muller games, the exact complexity has not been fully
investigated. In Section 2.3 we show that the complexity of deciding Muller games is
P SPACE -complete for many types of presentation.
We also establish a framework in which the expressiveness and succinctness of
different types of winning conditions can be compared. We introduce a notion of
polynomial time translatability between formalisms which gives rise to a notion of
game complexity stronger than that implied by polynomial time reductions of the corresponding decision problems. Informally, a specification is translatable into another if
the representation of a game in the first can be transformed into a representation of the
same game in the second.
The complexity results we establish for Muller games allow us to show two important problems related to Muller automata are also P SPACE -complete: the nonemptiness problem and the model-checking problem on regular trees.
The chapter is organised as follows. In Section 2.1 we present the formal definitions
of arenas, games and strategies that we use throughout the remainder of the dissertation.
In Section 2.2 we introduce the notion of a winning condition type, a formalization for
specifying winning conditions. We provide examples from the literature and we consider the notion of translatability between condition types. In Section 2.3 we present
some results regarding the complexity of deciding the games we consider here, including the P SPACE -completeness result for Muller games, and a co-NP-completeness result for two games we introduce. Finally, in Section 2.4 we show that the non-emptiness
and model checking problems for Muller tree automata are also P SPACE -complete.
2.1 Preliminaries
In this section we present the definitions of arenas, games and strategies that we use
throughout the dissertation. The definitions we use follow [GTW02]. In Section 2.1.4
we introduce a generalization of bisimulation appropriate for arenas and games, game
simulation, and we show how it can be used to translate plays and strategies from one
arena to another.
2.1.1 Arenas
Our first definition is a generalization of a transition system where two entities or players control the transitions.
Definition 2.1 (Arena). An arena is a tuple A := (V, V0 , V1 , E, vI ) where:
2.1. PRELIMINARIES
13
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)
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Figure 2.1: An example of an arena
• (V, E) is a directed graph,
• V0 , the set of Player 0 vertices, and V1 , the set of Player 1 vertices, form a
partition of V , and
• vI ∈ V is the initial vertex.
Viewing arenas as directed graphs with some additional structure, we define the
notions of subarena and induced subarena in the obvious way. Figure 2.1 illustrates an
arena A with V0 (A) = {v4 , v5 , v6 } and V1 (A) = {v1 , v2 , v3 , v7 , v8 , v9 }.
Given an arena, A, we consider the following set of interactions between two players: Player 0 and Player 1.1 A token, or pebble, is placed on vI (A). Whenever the
pebble is on a vertex v ∈ V0 (A), Player 0 chooses a successor of v and moves the
pebble to that vertex, and similarly when the pebble is on a vertex v ∈ V1 (A), Player 1
chooses the move. This results in a (possibly infinite) sequence of vertices visited by
the pebble. We call such a sequence a play. More formally,
Definition 2.2 (Play). Given an arena A and v ∈ V (A), a play in A (from v) is a
(possibly infinite) sequence of vertices v1 v2 · · · such that v1 = v and for all i ≥ 1,
(vi , vi+1 ) ∈ E(A). If v is not specified, we assume the play is from vI (A). The set of
all plays in A from vI (A) is denoted by Plays(A).
We observe that if A0 is a subarena of A then Plays(A0 ) ⊆ Plays(A).
As an example, the infinite sequence v1 v4 v7 v5 v8 v6 v9 v4 v7 (v5 v2 )ω is a play in the
arena pictured in Figure 2.1, as is the finite sequence v1 v4 v7 v5 v8 v6 v9 v4 .
When one of the players has no choice of move, we may assume that there is only
one player as there is no meaningful interaction between the players.
Definition 2.3 (Single-player arena). Let A = (V, V0 , V1 , E, vI ) be an arena. We say
A is a single-player arena if for some i ∈ {0, 1} and every v ∈ Vi , dout (v) ≤ 1.
An important concept relating to arenas and the games we consider is the notion of
duality. In the dual situation, we interchange the roles of Player 0 and Player 1. This
gives us the following definition of a dual arena.
1 For
convenience we use the feminine pronoun for Player 0 and the masculine pronoun for Player 1
14
CHAPTER 2. INFINITE GAMES
Definition 2.4 (Dual arena). Let A = (V, V0 , V1 , E, vI ) be an arena. The dual arena
of A is the arena defined by Ae := (V, V1 , V0 , E, vI ).
e
We observe that for each arena A, Plays(A) = Plays(A).
2.1.2 Games
Arenas and plays establish the interactions that we are concerned with. We now use
these to define games by imposing outcomes for plays. The games we are interested
in are zero-sum games, that is, if one player wins then the other player loses. We can
therefore define a winning condition as a set of plays that are winning for one player,
say Player 0, working on the premise that if a play is not in that set then it is winning
for Player 1.
Definition 2.5 (Game). A game is a pair G := (A, Win) where A is an arena and
Win ⊆ Plays(A). For π ∈ Plays(A) if π ∈ Win, we say π is winning for Player 0,
otherwise π is winning for Player 1. A single-player game is a game (A, Win) where
A is a single player arena.
As we mentioned earlier, to consider algorithmic aspects of these games we need
to assume that they can be finitely presented. Muller games are an important example
of a class of finitely presentable games. With a Muller game, if a player cannot move
then he or she loses, otherwise the outcome of an infinite play is dependent on the set
of vertices visited infinitely often.
Definition 2.6 (Muller game). A game G = (A, Win) is a Muller game if A is finite
and there exists F ⊆ P(V (A)) such that for all π ∈ Plays(A):
(
π is finite and ends with a vertex from V1 (A), or
π ∈ Win ⇐⇒
π is infinite and {v : v occurs infinitely often in π} ∈ F.
If G is a Muller game, witnessed by F ⊆ P(V (A)), we write G = (A, F ).
As an example, consider the arena A pictured in Figure 2.1. Let F = {v2 , v5 } .
Then G = (A, F ) is a Muller game. The play v1 v4 v7 v5 v8 v6 v9 v4 v7 (v5 v2 )ω is winning
for Player 0, but the play v1 v4 v7 (v5 v8 v6 v9 v4 v7 )ω is winning for Player 1.
The games used in the literature in the study of logics and automata are generally
Muller games. In these games, the set F is often not explicitly given but is specified
by means of a condition. Different types of condition lead to various different types of
games. We explore this in more detail in Section 2.2.
An important subclass of Muller games are the games where only one player wins
any infinite play. Games such as Ehrenfeucht-Fraı̈ssé games (on finite structures) [EF99]
and the graph searching games we consider in Chapter 5 are examples of these types
of games.
Definition 2.7 (Simple game). A Muller game G = (A, F ) is a simple game if either
F = ∅, or F = P(V (A)).
2.1. PRELIMINARIES
15
Two other important subclasses of Muller games which we consider in this chapter
are union-closed and upward-closed games.
Definition 2.8 (Union-closed and Upward-closed games). A Muller game G = (A, F )
is union-closed if for all X, Y ∈ F, X ∪ Y ∈ F. G is upward-closed if for all X ∈ F
and Y ⊇ X, Y ∈ F.
Remark. Union-closed games are often called Streett-Rabin games in the literature, as
Player 0’s winning set can be specified by a set of Streett pairs (see Definition 2.38
below) and Player 1’s winning set can be specified by a set of Rabin pairs (see Definition 2.37). However, to minimize confusion, we reserve the term Streett game for
union-closed games with a condition presented as a set of Streett pairs, and the term
Rabin game for the dual of a union-closed game (see below) with a condition presented
as a set of Rabin pairs.
We conclude this section by considering dual games and subgames. In Definition 2.4 we defined the dual of an arena. The dual game is played on the dual arena, but
we have to complement the winning condition in order to fully interchange the roles of
the players. That is,
e := (A,
e Win)
Definition 2.9 (Dual game). Let G = (A, Win) be a game. The game G
where Ae is the dual arena of A and Win = Plays(A) \ Win is the dual game of G.
Given a game on an arena A we can define a restricted game on a subarena A0 by
restricting the winning condition to valid plays in the subarena.
Definition 2.10 (Subgame). Let G = (A, Win) be a game, and A0 a subarena of A.
The subgame induced by A0 is the game G0 = (A0 , Win0 ) where Win0 = Win ∩
Plays(A0 ).
2.1.3 Strategies
As with most games we are less interested in outcomes of single plays in the game and
more interested in the existence of strategies that ensure one player wins against any
choice of moves from the other player.
Definition 2.11 (Strategy). Let A = (V, V0 , V1 , E, vI ) be an arena. A strategy (for
Player i) in A is a partial function σ : V ∗ Vi → V such that if σ(v1 v2 · · · vn ) = v 0 then
(vn , v 0 ) ∈ E. A play π = v1 v2 · · · is consistent with a strategy σ if for all j < |π|
such that vj ∈ Vi , σ(v1 v2 · · · vj ) = vj+1 .
Given a sequence of vertices visited, ending with a vertex in Vi , a strategy for
Player i gives the vertex that Player i should then play to. We observe that given a
strategy σ for Player 0 and a strategy τ for Player 1 from any vertex v there is a unique
maximal play πτσ from v consistent with σ and τ in the sense that any play consistent
with both strategies is a prefix of πτσ . We call this play the play (from v) defined by
strategies σ and τ .
A useful class of strategies are those that can be defined from a fixed number of
previously visited vertices.
16
CHAPTER 2. INFINITE GAMES
Definition 2.12 (Strategy memory). If a strategy σ has the property that for some fixed
m, σ(w) = σ(w0 ) if w and w0 agree on their last m letters, then we say that the strategy
requires finite memory (of size m − 1). If m = 1, we say the strategy is memoryless or
positional.
Strategies extend to games in the obvious way.
Definition 2.13 (Game strategies). Given a game G = (A, Win), a strategy for Player i
in G is a strategy for Player i in A. A strategy σ for Player i is winning if all plays
consistent with σ are winning for Player i. Player i wins G if Player i has a winning
strategy from vI (A).
We observe that for any play π = v1 v2 · · · vn in a Muller game, consistent with a
winning strategy σ for Player i, if vn ∈ Vi (A) then σ(π) is defined.
Earlier we alluded to the following important result of Büchi and Landweber [BL69].
Theorem 2.14 ([BL69]). Let G = (A, F ) be a Muller game. One player has a winning
strategy on G with finite memory of size at most |V (A)|!.
An immediate corollary of this is that Muller games are decidable: we can check
all possible strategies for both players that use at most |V (A)|! memory, and see if the
corresponding defined plays are winning. However, the complexity bounds on such
an algorithm are enormous. In [McN93] McNaughton provided an algorithm with
considerably better space and time bounds.
Theorem 2.15 ([McN93]). Let G = (A, F ) be a Muller game with A = (V, V0 , V1 , E, vI ).
Whether Player 0 has a winning strategy from vI can be decided in time O(|V |2 |E||V |!)
and space O(|V |2 ).
For union-closed games and their duals we can reduce the memory requirement for
a winning strategy.
Theorem 2.16 ([Kla94]). Let G = (A, F ) be a Muller game. If F is closed under
unions and Player 1 has a winning strategy, then Player 1 has a memoryless winning
strategy. Dually, if the complement of F is closed under union and Player 0 has a
winning strategy, then Player 0 has a memoryless winning strategy.
Two useful tools for constructing decidability algorithms are force-sets and avoid-sets.
Definition 2.17 (Force-set and Avoid-set). Let A be an arena, and X, Y ⊆ V (A).
i
The set ForceX
(Y ) is the set of vertices from which Player i has a strategy σ such
that any play consistent with σ reaches some vertex in Y without leaving X. The set
i
(Y ) is the set of vertices from which Player i has a strategy σ such that any
AvoidX
play consistent with σ that remains in X avoids all vertices in Y .
1−i
i
We observe from the definitions that ForceX
(Y ) = X \ AvoidX
(Y ). We also
observe that we may assume the strategies σ are memoryless: if Player i can force the
play from v to some vertex of Y , the play to v is irrelevant.
Computing a force-set is an instance of the well-known alternating reachability
problem, and in Algorithm 2.1 we present the standard algorithm for computing a
force-set. Nerode, Remmel and Yakhnis [NRY96] provide an implementation of this
algorithm which runs in time O(|E(A)|), giving us the following:
2.1. PRELIMINARIES
17
0
Lemma 2.18. Let A be an arena. For any sets X, Y ⊆ V (A), ForceX
(Y ) can be
computed in time O(|E(A)|)
0
Algorithm 2.1 F ORCE X
(Y )
Returns: The set of vertices v ∈ V (A) such that Player 0 has a strategy to force a
play from v to some element of Y without visiting a vertex outside X.
let R = {v ∈ V0 (A) ∩ X : there exists w ∈ Y with (v, w) ∈ E(A)}.
let S = {v ∈ V1 (A) ∩ X : for all w with (v, w) ∈ E(A), w ∈ Y }.
if R ∪ S ⊆ Y then
return Y
else
0
return F ORCE X
(R ∪ S ∪ Y ).
2.1.4 Simulations
One of the most important concepts in transition systems is the notion of bisimulation.
Two transition systems are bisimilar if each system can simulate the other. That is,
Definition 2.19 (Bisimulation). Let T = (S, s0 , E) and T 0 = (S 0 , s00 , E 0 ) be transition
systems. We say T and T 0 are bisimilar if there exists a relation ∼⊆ S × S 0 such that:
• s0 ∼ s00 ,
• If (s, t) ∈ E and s ∼ s0 then there exists t0 ∈ S 0 such that (s0 , t0 ) ∈ E 0 and
t ∼ t0 , and
• If (s0 , t0 ) ∈ E 0 and s ∼ s0 then there exists t ∈ S such that (s, t) ∈ E and t ∼ t0 .
We now consider a generalization of bisimulation appropriate for arenas.
Definition 2.20 (Game simulation). Let A and A0 be arenas. A game
simulation from
A to A0 is a relation S⊆ V0 (A) × V0 (A0 ) ∪ V1 (A) × V1 (A0 ) such that:
(SIM-1) vI (A) S vI (A0 ),
(SIM-2) If (u, v) ∈ E(A), u ∈ V0 (A) and u S u0 , then there exists v 0 ∈ V (A0 ) such
that (u0 , v 0 ) ∈ E(A0 ) and v S v 0 , and
(SIM-3) If (u0 , v 0 ) ∈ E(A0 ), u0 ∈ V1 (A0 ) and u S u0 , then there exists v ∈ V (A)
such that (u, v) ∈ E(A) and v S v 0 .
We write A - A0 if there exists a game simulation from A to A0 .
f0 - A.
e In
We observe that - is reflexive and transitive and if A - A0 then A
Proposition 2.28 we show that it is also antisymmetric (up to bisimulation).
If A - A0 , then Player 0 can simulate plays on A0 as plays on A: every move made
by Player 1 on A0 can be translated to a move on A, and for every response of Player 0
in A, there is a corresponding response on A0 . Dually, Player 1 can simulate a play on
A as a play on A0 . More precisely,
18
CHAPTER 2. INFINITE GAMES
Lemma 2.21. Let A and A0 be arenas, and let S be a simulation from A to A0 . For
any strategy σ for Player 0 in A, and any strategy τ 0 for Player 1 in A0 , there exists
a strategy σ 0 for Player 0 in A0 and a strategy τ for Player 1 in A such that if π =
v0 v1 · · · ∈ Plays(A) is a play from v0 = vI (A) consistent with σ and τ and π 0 =
v00 v10 · · · ∈ Plays(A0 ) is a play from v00 = vI (A0 ) consistent with σ 0 and τ 0 , then
vi S vi0 for all i, 0 ≤ i ≤ min{|π|, |π 0 |}.
Proof. We define σ 0 and τ as follows. Let π = v0 v1 v2 · · · vn and π 0 = v00 v10 · · · vn0 and
suppose vi S vi0 for all i, 0 ≤ i ≤ n. Suppose first that vn ∈ V0 (A) (so vn0 ∈ V0 (A0 ))
and σ(π) = vn+1 . Since (vn , vn+1 ) ∈ E(A) and vn S vn0 , from Condition (SIM-2)
0
0
0
there exists vn+1
such that (vn0 , vn+1
) ∈ E(A0 ) and vn+1 S vn+1
. Define σ 0 (π 0 ) :=
0
0
0
0 0
0
vn+1 . Now suppose vn ∈ V1 (A) (so vn ∈ V1 (A )). Let τ (π ) = vn+1 and let vn+1 be
0
the successor of vn , such that vn+1 S vn+1
guaranteed by Condition (SIM-3). Define
τ (π) = vn+1 . We observe that although σ 0 and τ are only defined for some plays,
this definition is sufficient: as v0 S v00 , it follows by induction that for every play
π 0 = v00 v10 · · · vn0 consistent with σ 0 and τ 0 there is a play π = v0 v1 · · · vn (consistent
with σ) such that vi S vi0 for all i, 0 ≤ i ≤ n. Thus if vn0 ∈ V0 (A0 ), σ(π 0 ) is welldefined.
u
t
We observe that the strategies σ 0 and τ are independently derivable from τ 0 and σ
respectively. That is, we can interchange the ∀τ 0 and ∃σ 0 (or the ∀σ and ∃τ ) quantifications to obtain:
Corollary 2.22. Let A and A0 be arenas, and let S be a game simulation from A to A0 .
For every strategy σ for Player 0 in A there exists a strategy σ 0 for Player 0 in A0 such
that for every play v00 v10 · · · consistent with σ 0 there exists a play v0 v1 · · · , consistent
with σ such that vi S vi0 for all i. Dually, for every strategy τ 0 for Player 1 in A0 there
exists a strategy τ for Player 1 in A such that for every play v0 v1 · · · consistent with τ
there exists a play v0 v1 · · · , consistent with τ 0 such that vi S vi0 for all i.
We call the strategies which we can derive in such a manner simulated strategies.
Definition 2.23 (Simulated search strategy). Let A, A0 , S, σ, σ 0 , τ and τ 0 be as above.
We call σ 0 a S-simulated strategy of σ, and τ a S-simulated strategy of τ 0 .
We can use game simulations to translate winning strategies from one game into
winning strategies in another. However, we require that a simulation respects the winning condition in some sense.
Definition 2.24 (Faithful simulation). Let G = (A, Win) and G0 = (A0 , Win0 ) be
games. Let S be a game simulation from A to A0 , and let S also denote the pointwise
extension of the relation to plays: π S π 0 if |π| = |π 0 | and vi S vi0 for all vi ∈ π
and vi0 ∈ π 0 . We say S is (Win, Win0 )-faithful if for all π ∈ Plays(A) and all π 0 ∈
Plays(A0 ) such that π S π 0 :
π ∈ Win =⇒ π 0 ∈ Win0 .
The next result follows immediately from the definitions.
2.2. WINNING CONDITION PRESENTATIONS
19
Proposition 2.25. Let G = (A, Win) and G0 = (A0 , Win0 ) be games. Let S be a
(Win, Win0 )-faithful game simulation from A to A0 . If σ is a winning strategy for
Player 0 in G then any S-simulated strategy is a winning strategy for Player 0 in G0 .
Dually, if τ 0 is a winning strategy for Player 1 in G0 then any S-simulated strategy is a
winning strategy for Player 1 in G.
For simple games checking if a game simulation is faithful is relatively easy. It
follows from the definition of a game simulation that all finite plays automatically
satisfy the criterion. Thus it suffices to check the infinite plays. But for simple games
these are vacuously satisfied in two cases:
Lemma 2.26. Let G = (A, F ) and G0 = (A0 , F 0 ) be Muller games and let S be a
simulation from A to A0 . If either F = ∅ or F 0 = P(V (A0 )) then S is faithful.
Corollary 2.27. Let G = (A, F ) and G0 = (A0 , F 0 ) be Muller games such that F = ∅
or F 0 = P(V (A0 )). If A - A0 and Player 0 wins G, then Player 0 wins G0 . Dually, if
A - A0 and Player 1 wins G0 , then Player 1 wins G.
We conclude this section by showing how game simulations relate to bisimulation.
Proposition 2.28. Let A = (V, V0 , V1 , E, vI ) and A0 = (V 0 , V00 , V10 , E 0 , vI0 ) be arenas.
If A - A0 and A0 - A then the transition systems (V, vI , E) and (V 0 , vI0 , E 0 ) are
bisimilar.
Proof. Let S be a game simulation from A to A0 and let S0 be a game simulation from
A0 to A. It follows from the definitions that the relation S ∪(S0 )−1 is a bisimulation
between the two transition systems.
u
t
2.2 Winning condition presentations
As we discussed above, if we are interested in investigating the complexity of the problem of deciding Muller games, we need to consider the manner in which the winning
condition is presented. As we see in Section 2.2.1, for many games that occur in the
literature relating to logics and automata the winning condition can be expressed in a
more efficient manner than simply listing the elements of F . To formally describe such
specifications, we introduce the concept of a condition type.
Definition 2.29 (Condition type). A condition type is a function A which maps an arena
A to a pair (I A , |=A ) where I A is a set and |=A ⊆ Plays(A) × I A is the acceptance
relation. We call elements of I A condition types (or simply, conditions). A regular
condition type maps an arena A to a pair (I A , |=A ) where I A is a set of conditions
and |=A ⊆ P(V (A)) × I A .
Remark. In the sequel we will generally regard the relation |=A as intrinsically defined,
and associate A(A) with the set I A . That is, we will use Ω ∈ A(A) to indicate Ω ∈ I A .
A (regular) condition type defines a family of (Muller) games in the following manner.
Let A be a condition type, A an arena, and A(A) = (I A , |=A ). For Ω ∈ I A , the
game (A, Ω) is the game (A, Win) where Win = {π ∈ Plays(A) : π |=A Ω}. We
CHAPTER 2. INFINITE GAMES
20
generally call a game where the winning condition is specified by a condition of type
A an A-game, for example a parity game is a game where the winning condition is
specified by a parity condition (see Definition 2.41 below). We can now state precisely
the decision problem we are interested in.
A-G AME
Instance:
Problem:
A game G = (A, Ω) where Ω ∈ A(A).
Does Player 0 have a winning strategy in G?
The exploration of the complexity of this problem is one of the main research problems that this dissertation addresses.
Research aim. Investigate the complexity of deciding A-G AME for various (regular)
condition types A.
2.2.1 Examples
We now give some examples of regular condition types that occur in the literature. First
we observe that an instance Ω ∈ A(A) of a regular condition type A defines a family
of subsets of V (A):
FΩ := {I ⊆ V (A) : I |=A Ω}.
We call this the set specified by the condition Ω. In the examples below, we describe
the set specified by a condition to define the acceptance relation |=A .
General purpose condition types
The first examples we consider are general purpose formalisms in that they may be
used to specify any family of sets.
The most straightforward presentation of the winning condition of a Muller game
(A, F ) is given by explicitly listing all elements of F . We call this an explicit presentation. We can view such a formalism in our framework as follows:
Definition 2.30 (Explicit condition type). An instance of the explicit condition type
is a set F ⊆ P(V (A)). The set specified by an instance is the set which defines the
instance.
In the literature an explicit presentation is sometimes called a Muller condition.
However, we reserve that term for the more commonly used presentation for Muller
games in terms of colours given next.
Definition 2.31 (Muller condition type). An instance of the Muller condition type is
a pair (χ, C) where, for some set C, χ : V (A) → C and C ⊆ P(C). The set F(χ,C)
specified by a Muller condition (χ, C) is the set {I ⊆ V (A) : χ(I) ∈ C}.
To distinguish Muller games from games with a winning condition specified by
a Muller condition, we explicitly state the nature of the presentation of the winning
condition if it is critical.
2.2. WINNING CONDITION PRESENTATIONS
21
From a more practical perspective, when considering applications of these types of
games it may be the case that there are vertices whose appearance in any infinite run is
irrelevant. This leads to the definition of a win-set condition.
Definition 2.32 (Win-set condition type). An instance of the win-set condition type is
a pair (W, W) where W ⊆ V (A) and W ⊆ P(W ). The set F(W,W) specified by a
win-set condition (W, W) is the set {I ⊆ V (A) : W ∩ I ∈ W}.
Another way to describe a winning condition is as a boolean formula. Such a formalism is somewhat closer in nature than the specifications we have so far considered
to the motivating problem of verifying reactive systems: requirements of such systems
are more readily expressed as logical formulas. Winning conditions of this kind were
considered by Emerson and Lei [EL85].
Definition 2.33 (Emerson-Lei condition type). An instance of the Emerson-Lei condition type is a boolean formula ϕ with variables from the set V (A). The set Fϕ specified
by an Emerson-Lei condition ϕ is the collection of sets I ⊆ V (A) such that the truth
assignment that maps each element of I to true and each element of V (A) \ I to false
satisfies ϕ.
A boolean formula can contain a lot of repetition, so it may be more efficient to
consider boolean circuits rather than formulas. This motivates one of the most succinct
types of winning condition we consider.
Definition 2.34 (Circuit condition type). An instance of the circuit condition type is a
boolean circuit C with input nodes from the set V (A) and one output node. The set
FC specified by a circuit condition C is the collection of sets I ⊆ V (A) such that C
outputs true when each input corresponding to a vertex in I is set to true and all other
inputs are set to false.
The final general purpose formalisms we consider are somewhat more exotic. In [Zie98],
Zielonka introduced a representation for a family of subsets of a set V , F ⊆ P(V ), in
terms of a labelled tree where the labels on the nodes are subsets of V .
Definition 2.35 (Zielonka tree and Zielonka DAG). Let V be a set and F ⊆ P(V ).
The Zielonka tree (also called a split tree of the set F , ZF ,V , is defined inductively as:
1. If V ∈
/ F then ZF ,V = ZF,V , where F = P(V ) \ F .
2. If V ∈ F then the root of ZF ,V is labelled with V . Let M1 , M2 , . . . , Mk be the
⊆-maximal sets in F , and let F |Mi = F ∩ P(Mi ). The successors of the root
are the subtrees ZF |Mi ,Mi , for 1 ≤ i ≤ k.
A Zielonka DAG is constructed as a Zielonka tree except nodes labelled by the same set
are identified, making it a directed acyclic graph. Nodes of ZF ,V labelled by elements
of F are called 0-level nodes, and other nodes are 1-level nodes.
Zielonka trees are intimately related to Muller games. In particular they identify
the size of memory required for a winning strategy: the “amount” of branching of
22
CHAPTER 2. INFINITE GAMES
0-level nodes indicates the maximum amount of memory required for a winning strategy for Player 0, and similarly for 1-level nodes and Player 1 [DJW97]. For example,
the 1-level nodes of a Zielonka tree of a union-closed family of sets have at most one
successor, indicating that if Player 1 has a winning strategy then he has a memoryless winning strategy. Thus we also consider games where the winning condition is
specified as a Zielonka tree (or the more succinct Zielonka DAG).
Definition 2.36 (Zielonka tree and Zielonka DAG condition types). An instance of
the Zielonka tree (DAG) condition type is a Zielonka tree (DAG) ZF ,V (A) for some
F ⊆ P(V (A)). The set specified by an instance is the set F used to define the instance.
Other condition types
We now consider formalisms that can only specify restricted families of sets such as
union-closed or upward-closed families. The first formalism we consider is a wellknown specification, introduced by Rabin in [Rab72] as an acceptance condition for
infinite automata.
Definition 2.37 (Rabin condition type). An instance of the Rabin condition type is a
set of pairs Ω = {(Li , Ri ) : 1 ≤ i ≤ m}. The set FΩ specified by a Rabin condition
Ω is the collection of sets I ⊆ V (A) such that there exists an i, 1 ≤ i ≤ m, such that
I ∩ Li 6= ∅ and I ∩ Ri = ∅.
The remaining formalisms we consider can only be used to specify families of sets
that are closed under union. The first of these, the Streett condition type, introduced
in [Str82], is similar to the Rabin condition type.
Definition 2.38 (Streett condition type). An instance of the Streett condition type is a
set of pairs Ω = {(Li , Ri ) : 1 ≤ i ≤ m}. The set FΩ specified by a Streett condition
Ω is the collection of sets I ⊆ V (A) such that for all i, 1 ≤ i ≤ m, either I ∩ Li 6= ∅
or I ∩ Ri = ∅.
The Streett condition type is useful for describing fairness conditions such as those
considered in [EL85]. An example of a fairness condition for infinite computations is:
“every process enabled infinitely often is executed infinitely often”. Viewing vertices of
an arena as states of an infinite computation system where some processes are executed
and some are enabled, this is equivalent to saying “for every process, either the set of
states which enable the process is visited finitely often or the set of states which execute
the process is visited infinitely often”, which we see is easily interpreted as a Streett
condition.
The Streett and Rabin condition types are dual in the following sense: for any set
F ⊆ P(V (A)) which can be specified by a Streett condition, there is a Rabin condition
which specifies P(V (A)) \ F , and conversely. Indeed, if Ω = {(Li , Ri ) : 1 ≤ i ≤ m}
e = {(Ri , Li ) : 1 ≤ i ≤ m}
is a Streett condition, then for the Rabin condition Ω
we have FΩ
e = P(V (A)) \ FΩ . This implies that the dual of a Streett game can be
expressed as a Rabin game, and conversely the dual of a Rabin game can be expressed
as a Streett game.
If we are interested in specifying union-closed families of sets efficiently, we can
consider the closure under union of a given set. This motivates the following definition:
2.2. WINNING CONDITION PRESENTATIONS
23
Definition 2.39 (Basis condition type). An instance of the basis condition type is a set
B ⊆ P(V (A)). The set FB specified by a basis condition
S B is the collection of sets
I ⊆ V (A) such that there are B1 , . . . , Bn ∈ B with I = 1≤i≤n Bi .
In a similar manner to the basis condition type, if we are interested in efficiently
specifying an upward-closed family of sets, we can explicitly list the ⊆-minimal elements of the family. This gives us the superset condition type, also called a superset
Muller condition in [LTMN02].
Definition 2.40 (Superset condition type). An instance of the superset condition type
is a set M ⊆ P(V (A)). The set FM specified by a superset condition M is the set
{I ⊆ V (A) : M ⊆ I for some M ∈ M}.
The final formalism we consider is one of the most important and interesting Muller
condition types, the parity condition type.
Definition 2.41 (Parity condition type). An instance of the parity condition type is a
function χ : V (A) → P where P ⊆ ω is a set of priorities. The set Fχ specified by a
parity condition χ is the collection of sets I ⊆ V (A) such that max{χ(v) : v ∈ I} is
even.
Remark. We have technically defined here the max-parity condition. There is an equivalent formalism sometimes considered where the parity of the minimum priority visited
infinitely often determines the winner, called the min-parity condition. Throughout this
dissertation we only consider the max-parity condition.
It is not difficult to show that the set specified by a parity condition is closed under
union as is the complement of the set specified. Therefore, from Theorem 2.16 we have
the following:
Theorem 2.42 (Memoryless determinacy of parity games [EJ91, Mos91]). Let G =
(A, χ) be a parity game. The player with a winning strategy has a winning strategy
which is memoryless.
Indeed, any union-closed set with a union-closed complement can be specified by
a parity condition, implying that the parity condition is one of the most expressive
conditions where memoryless strategies are sufficient for both players. This result is
very useful in the study of infinite games and automata: one approach to showing
that Muller automata are closed under complementation is to reduce the problem to
a parity game, and utilise the fact that if Player 1 has a winning strategy then he has
a memoryless strategy to construct an automaton which accepts the complementary
language [EJ91].
One of the reasons why parity games are an interesting class of games to study is
that the exact complexity of the problem of deciding the winner remains elusive. In
Chapter 3 we discuss this and other reasons why parity games are important in more
detail.
CHAPTER 2. INFINITE GAMES
24
2.2.2 Translations
We now present a framework in which we can compare the expressiveness and succinctness of condition types by considering transformations between games which keep
the arena the same. More precisely, we define what it means for a condition type to be
translatable to another condition type as follows.
Definition 2.43 (Translatable). Given two condition types A and B, we say that A
A
is polynomially translatable to B if for any arena A, with A(A) = (IA
, |=A
A ) and
A
A
A
A
A
B(A) = (IB , |=B ), there is a function f : IA → IB such that for all Ω ∈ IA
:
• f (Ω) is computed in time polynomial in |A| + |Ω|, and
A
• For all π ∈ Plays(A), π |=A
A Ω ⇐⇒ π |=B f (Ω).
As we are only interested in polynomial translations, we simply say A is translatable to B to mean that it is polynomially translatable. Clearly, if condition type A
is translatable to B then the problem of deciding the winner for games of type A is
reducible in polynomial time to the corresponding problem for games of type B. That
is,
Lemma 2.44. Let A and B be condition types such that A is translatable to B. Then
there is a polynomial time reduction from A-G AME to B-G AME.
If condition type A is not translatable to B this may be for one of three reasons.
Either A is more expressive than B in that there are sets F that can be expressed using
conditions from A but no condition from B can specify F ; or there are some sets
for which the representation of type A is necessarily more succinct; or the translation,
while not size-increasing, can not be computed in polynomial time. We are primarily
interested in the second situation. Formally, we say
Definition 2.45 (Succinctness). A is more succinct than B if B is translatable to A but
A is not translatable to B.
We now consider translations between some of the condition types we defined in
Section 2.2.1.
Translations between general purpose condition types
It is straightforward to show that win-set conditions are more succinct than explicit
presentations. To translate an explicitly presented game (A, F ) to a win-set condition,
simply take W = V (A) and W = F . To show that win-set conditions are not translatable to explicit presentations, consider a game where W = ∅ and W = {∅}. The set
F(W,W) specified by this condition consists of all subsets of V (A) and thus an explicit
presentation must be exponential in length.
Proposition 2.46. The win-set condition type is more succinct than an explicit presentation.
2.2. WINNING CONDITION PRESENTATIONS
25
Similarly, there is a trivial translation from the Emerson-Lei condition type to the
circuit condition type. However, the question of whether there is a translation in the
other direction is an important open problem in the field of circuit complexity [Pap95].
Open problem 2.47. Is the circuit condition type more succinct than the Emerson-Lei
condition type?
We now show, through the next theorems, that circuit presentations are more succinct than Zielonka DAG presentations, which, along with Emerson-Lei presentations,
are more succinct than Muller presentations, which are in turn more succinct than winset presentations.
Theorem 2.48. The Muller condition type is more succinct than the win-set condition
type.
Proof. Given a win-set game A, (W, W) , we construct a Muller condition describing
the same set of subsets as (W, W). For the set of colours we use C = W ∪ {c}, where
c is distinct from any element of W . The colouring function χ : V (A) → C is then
defined as:
• χ(w) = w for w ∈ W ,
• χ(v) = c for v ∈
/ W.
The family C of subsets of C is the set X, X ∪ {c} : X ∈ W . For I ⊆ V , if I ⊆ W ,
then χ(I) = I otherwise χ(I) = {c} ∪ I. Either way, I ∩ W is in W if, and only if,
χ(I) ∈ C.
To show that there is no translation in the other direction, consider a Muller game
on A, where half of V (A), Vr , iscoloured
red, the other half coloured blue, and the
family of sets of colours is C = {red} . The family F described by this condition
consists of the 2|V (A)|/2 − 1 non-empty subsets of Vr . Now consider trying to describe
this family using a win-set condition. In general, for the set F 0 specified by the win-set
condition (W, W), any v ∈
/ W , and X ⊆ V (A) we have {v} ∪ X ∈ F 0 ⇔ X ∈ F 0 .
Observe that in our game no vertex has this latter property: if v ∈ Vr , then {v} ∈ F,
but ∅ ∈
/ F; and if v ∈
/ Vr then {v} ∪ Vr ∈
/ F, but Vr ∈ F. Thus our win-set, W must
be equal to V (A), and W is the explicit listing of the 2|V (A)|/2 − 1 subsets of Vr . Thus
(W, W) cannot be produced in polynomial time.
u
t
Theorem 2.49. The Zielonka DAG condition type is more succinct than the Muller
condition type.
Proof. Given a Muller game consisting of an arena A = (V, V0 , V1 , E, vI ), a colouring
χ : V → C and a family C of subsets of C, we construct a Zielonka DAG ZF ,V which
describes the same set of subsets of V (A) as the Muller condition (χ, C). Consider
the Zielonka DAG ZC,C , whose nodes are labelled by sets of colours. If we replace a
label L ⊆ C in this tree with the set {v ∈ V : χ(v) ∈ L} then we obtain a Zielonka
DAG ZF ,V over the set of vertices. We argue that F is, in fact, the set specified by
the Muller condition (χ, C) and then show that ZC,C can be constructed in polynomial
CHAPTER 2. INFINITE GAMES
26
time. Since the translation from ZC,C to ZF ,V involves an increase in size by at most
a factor of |V |, this establishes that Muller games are translatable to Zielonka DAGs.
Let I ⊆ V be a set of vertices. If I ∈ F then, by the definition of Zielonka DAGs,
I is a subset of a label X of a 0-level node t of ZF ,V and is not contained in any
of the labels of the 1-level successors of t. That is, for each 1-level successor u of t,
there is a vertex v ∈ I such that χ(v) 6∈ χ(Lu ) where Lu is the label of u. Moreover,
χ(I) ⊆ χ(X). Now χ(X) is, by construction, the label of a 0-level node of ZC,C and
we have established that χ(I) is contained in this label and is not contained in any of
the labels of the 1-level successors of that node. Therefore, χ(I) ∈ C. Similarly, by
interchanging 0-level and 1-level nodes, χ(I) ∈
/ C if I ∈
/ F.
To show that we can construct ZC,C in polynomial time, observe first that every
subset X ⊆ C has at most |C| maximal subsets. Note further that the label of any node
in ZC,C is either C, some element of C or a maximal (proper) subset of an element of
C. Thus, ZC,C is no larger than 1 + |C| + |C||C|. This bound on the size of the DAG
is easily turned into a bound on the time required to construct it, using the inductive
definition of Zielonka trees. Thus, we have shown that the Muller condition type is
translatable into the Zielonka DAG condition type.
To show there is no translation in the other direction, consider the family F of
subsets of V (A) which consist of 2 or more elements. The Zielonka DAG which
describes this family consists of |V (A)| + 1 nodes – one 0-level node labelled by
V (A), and |V (A)| 1-level nodes labelled by the singleton subsets of V (A). However,
to express this as a Muller condition, each vertex must have a distinct colour since
for any pair of vertices there is a set in F that contains one but not the other. Thus,
|C| = |F | = 2|V (A)| − |V (A)| − 1. It follows that the translation from Zielonka DAGs
to Muller conditions cannot be done in polynomial time.
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t
To show the remaining results, we use the following observation:
Lemma 2.50. There is no translation from the Emerson-Lei condition type to the
Zielonka DAG condition type.
Proof. Let V (A) = V = {x1 , . . . , x2k }, and consider the family of sets F described
by the formula
_
ϕ :=
(x2i−1 ∧ x2i ).
1≤i≤k
Clearly |ϕ| = O(|V (A)|). Now consider the Zielonka DAG ZF ,V describing F . As
V ∈ F, the root of ZF ,V is a 0-level node labelled by V . The maximal subsets of
V not in F are the 2k subsets containing exactly one of {x2i−1 , x2i } for 1 ≤ i ≤ k.
Thus ZF ,V must have at least this number of nodes, and is therefore not constructible
in polynomial time.
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Theorem 2.51. The Emerson-Lei condition type is more succinct than the Muller condition type.
Proof. Given a Muller game consisting of an arena A, a colouring χ : V (A) → C and
2.2. WINNING CONDITION PRESENTATIONS
27
a family C of subsets of C, let ϕ be the boolean formula defined as:
_^ _ ^ ^
¬v .
ϕ :=
v ∧
X∈C c∈X χ(v)=c
χ(v)=c
c∈X
/
It is easy to see that a subset I ⊆ V (A) satisfies ϕ if, and only if, there is some set
X ∈ C such that for all colours c ∈ X there is some v ∈ I such that χ(v) = c and
for all colours c0 ∈
/ X there is no v ∈ I such that χ(v) = c0 . Since ϕ can clearly be
constructed in time polynomial in |C| + |V (A)|, it follows that there is a translation
from the Muller condition type to the Emerson-Lei condition type.
For the reverse direction, we observe that as there is a translation from the Muller
condition type to the Zielonka DAG condition type, if there were a translation from
the Emerson-Lei condition type to the Muller condition type, this would contradict
Lemma 2.50 as “translatability” is transitive.
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Theorem 2.52. The circuit condition type is more succinct than the Zielonka DAG
condition type.
Proof. Given a Zielonka DAG game (A, ZF ,V ) where V = V (A), we define, for each
node t in ZF ,V a boolean circuit Ct . This circuit is defined by induction on the height
of t. For convenience, we associate each circuit with its output node. Suppose the label
of t is X. We have the following cases:
V
(i) t is a 0-level (X ∈ F) leaf: In this case, let Ct = x∈X
/ ¬x.
W
(ii) t is a 1-level (X ∈
/ F) leaf: In this case, let Ct = x∈X
/ x.
V
(iii) t is a 0-level node with k successors t1 , . . . , tk : In this case, let Ct = x∈X
/ ¬x ∧
Vk
.
C
i=1 ti
W
(iv) t is a 1-level node with k successors t1 , . . . , tk : In this case, let Ct = x∈X
/ x∨
Wk
i=1 Cti .
We claim that the condition F is specified by the circuit Cr where r is the root of ZF ,V .
This formula has size at most |V (A)||ZF ,V | and is constructed in polynomial time. To
show its correctness we argue by induction on the height of any node t with label X
that Ct defines the restriction of F to X. We consider the following cases:
(i) t is a 0-level leaf. In this case any subset of X is in F . I ⊆ V (A) satisfies Ct if,
and only if, no variable that is not in X appears in I, that is I ⊆ X.
(ii) t is a 1-level leaf. In this case any subset of X is not in F . Here I ⊆ V (A)
satisfies Ct if, and only if, there is some element in I which is not in X, that is
I 6⊆ X.
(iii) t is a 0-level node with k successors labelled by X1 , . . . , Xk . In this case any
subset of X is in F unless it is a subset of Xi for some i, in which case whether
it is in F is determined by nodes lower in the DAG. Here I ⊆ V (A) satisfies Ct
if, and only if, I is a subset of X and I satisfies Cti for all successors.
CHAPTER 2. INFINITE GAMES
28
Z IELONKA
C IRCUIT U
UU
iii
U
iiiii
E
MERSON -L EI
DAG
UUUUU
iii
UUU
iiii
M ULLER
W IN - SET
E XPLICIT
Figure 2.2: Summary of the succinctness results
(iv) t is a 1-level node with k successors labelled by X1 , . . . , Xk . In this case any
subset of X is not in F unless it is a subset of Xi for some i. Here I ⊆ V satisfies
Ct if, and only if, either I is not contained in X, or there is some successor ti
such that I satisfies Cti .
We observe that as there is a translation from the Emerson-Lei condition type to
the circuit condition type, Lemma 2.50 implies there is no translation from the circuit
condition type to the Zielonka DAG condition type.
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Figure 2.2 summarizes the succinctness results we have so far shown, with the more
succinct types towards the top. The dashed edge indicates that there is a translation but
it is not known whether there is a translation in the opposite direction.
Translations between union-closed condition types
Turning to union-closed condition types, we observe that the basis condition type is a
succinct way of describing union-closed sets. It is not even known if it is translatable
to the circuit condition type, the most succinct type considered above. In Section 2.3.2
we show that the problem of deciding basis games is co-NP-complete. It follows from
the NP-completeness of Rabin games [EJ88], and duality that the problem of deciding
Streett games is co-NP-complete. The following result implies that we cannot use
translatability to obtain upper or lower bounds on the complexity of basis games based
on the known bounds for Streett games.
Theorem 2.53. The basis and Streett condition types are incomparable with respect to
translatability. That is, neither is translatable to the other.
Proof. To show there is no translation from Streett games to basis games, let V (A) =
{x1 , .. . , x2k }, and considerthe Streett game with winning condition described by the
pairs (Li , ∅) : 1 ≤ i ≤ k , where
family of sets
Li = {x2i−1 , x2i }. Note that the
described by this condition is F = X ⊆ V (A) : ∀i X 6⊆ V (A) \ Li . Any basis for
F must include the minimal elements of F . However, the minimal elements include
M = {v1 , . . . , vk } : vi ∈ {x2i−1 , x2i } ,
2.2. WINNING CONDITION PRESENTATIONS
29
and |M| = 2k . Thus F cannot be represented by a basis constructible in polynomial
time.
To show there is no translation in the other direction, let V (A) = {x1 , . . . , x2k },
and consider the family F of sets formed by closing
B = {x2i−1 , x2i } : 1 ≤ i ≤ k
under union. Note that this is the same construction as for the proof of Theorem 2.52.
Observe that F contains 2k −1 sets, each with an even number of elements. Any Streett
condition which describes the same family must contain at least this number of pairs in
order to exclude the sets of odd cardinality. Thus F cannot be represented by a Streett
condition which is constructible in polynomial time.
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It should be clear that the superset condition type is translatable to the basis condition type. We include the result for completeness.
Proposition 2.54. The superset condition type is translatable to the basis condition
type.
We conclude these results with the following two observations regarding translations between explicit presentations and the basis and superset condition types.
Proposition 2.55. The superset condition type is more succinct than an explicit presentation of an upward-closed set.
Proof. Given an explicitly presented upward-closed game (A, F ), the set F , viewed as
a superset condition,
describes the same
set of subsets of V (A). Conversely, for
clearly
the superset game A, {v} : v ∈ V (A) , the set described by the winning condition
is of size 2|V (A)| − 1, and therefore cannot be explicitly presented in polynomial time.
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Corollary 2.56. The basis condition type is more succinct than an explicit presentation
of a union-closed set.
Proof. The fact that the basis condition type is not translatable to an explicit presentation follows from Proposition 2.55 and Proposition 2.54 as “translatable” is transitive.
The other direction is straightforward, the explicit presentation itself suffices as a basis.
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2.2.3 Extendibility
We now introduce a property of condition types that allows us to make simplifying
assumptions about the arena. We say a regular condition type is extendible if it can
“ignore” a set of added vertices. More precisely,
Definition 2.57 (Extendible condition type). Let A be a regular condition type. We
say A is extendible if for any arenas A and A0 such that V (A) ⊆ V (A0 ), and any
instance Ω ∈ A(A), there is an instance Ω0 ∈ A(A0 ), computable in time polynomial
in |Ω| + |V (A0 )|, such that FΩ0 = {I ⊆ V (A0 ) : I ∩ V (A) ∈ FΩ }.
30
CHAPTER 2. INFINITE GAMES
We observe that if |V (A0 )| − |V (A)| = m, then |FΩ0 | = 2m |FΩ |, so in particular,
an explicit presentation is not extendible. However, all the other condition types we
have so far considered are extendible.
Proposition 2.58. The following condition types are extendible: Muller, circuit, EmersonLei, Zielonka tree/DAG, win-set, parity, Rabin, Streett, basis, and superset.
Proof. Let us fix arenas A and A0 such that V (A) ⊆ V (A0 ). We show for each
condition type above how to compute the required instance Ω0 from a given Ω. It
follows from the definitions that for the circuit, Emerson-Lei, win-set, Rabin, Streett
and superset conditions taking Ω0 = Ω suffices. So let us consider the other condition
types.
Suppose Ω = (χ, C) is a Muller condition instance with χ : V (A) → C. We define
Ω0 = (χ0 , C 0 ) as follows. Let C 0 = C ∪ {c} where c is not an element of C. We define
(
χ(v) if v ∈ V (A)
0
χ (v) :=
c otherwise
and we define C 0 := C ∪ {I ∪ {c} : I ∈ C}. (χ0 , C 0 ) is clearly computable in time
polynomial in |Ω| + |V (A0 )|, and for every I ⊆ V (A0 ) we have χ0 (I) ∈ C 0 if, and only
if, χ(I ∩ V (A)) ∈ C. Thus Ω0 is as required.
Similarly, if Ω = (χ, P) is a parity condition, we let P0 = P ∪ {p} for some odd
p < min{χ(v) : v ∈ V (A)} and define χ0 (v) = p for v ∈
/ V (A), and χ(v) = v
otherwise. For any set I ⊆ V (A0 ), if I ∩ V (A) 6= ∅ then max{χ0 (v) : v ∈ I} =
max{χ(v) : v ∈ I ∩ V (A)}, so I ∈ FΩ0 if, and only if, I ∩ V (A) ∈ FΩ . Otherwise,
if I ∩ V (A) = ∅, then min{χ0 (v) : v ∈ I} = p, and as ∅ ∈
/ FΩ and p is odd, we have
/ FΩ . Thus Ω0 is as required.
I∈
/ FΩ0 and I ∩ V (A) ∈
Given a Zielonka structure ZF ,V where V = V (A), consider the Zielonka structure
Ω0 = ZF 0 ,V 0 , where V 0 = V (A0 ), defined by adding V (A0 )\ V (A) to each label. That
is, if t is a node in ZF ,V , labelled by X ⊆ V , then t is a node in ZF 0 ,V 0 labelled by
X ∪ (V (A0 ) \ V (A)). Now consider I ∈ F 0 . From the definition of a Zielonka
structure, I is a subset of a label of a 0-level node t and not a subset of a label of any
of the successors of t. Suppose t is labelled, in ZF ,V , by X, so I ⊆ X ∪ (V 0 \ V ).
Thus I ∩ V (A) ⊆ X. Now suppose I ∩ V (A) is a subset of Y , a label (in ZF ,V )
of a successor of t. It follows that I ⊆ Y ∪ (V 0 \ V ), and so I is a subset of a label
(in ZF 0 ,V 0 ) of a successor of t, contradicting the choice of t. So I ∩ V (A) ∈ F.
Interchanging the roles of 0-level nodes and 1-level nodes establishes that if I ∈
/ F0
0
then I ∩ V (A) ∈
/ F. Thus Ω is as required.
Finally, given an instance of a basis condition type Ω = B, we define Ω0 = B 0 as
follows:
B 0 = B ∪ {v} : v ∈ V (A0 ) \ V (A) .
Sn
Suppose I = i=1 Bi for sets B1 , . . . , Bn ∈ B 0 , where for some
Sm m ≤ n, Bi ∈ B for
i ≤ m. From the definition of B 0 , it follows that I ∩SV (A) = i=1 Bi , so I ∩ V (A) ∈
m
0
Bi . From the definition
FΩ . Conversely, if I ∩V (A) ∈ FΩ , let I ∩V (A) = i=1S
Snof B ,
n
0
there exists Bm+1 , . . . , Bn ∈ B such that I \ V (A) = i=m+1 Bi . So I = i=1 Bi
u
t
for B1 , . . . , Bn ∈ B 0 and hence I ∈ FΩ0 .
2.3. COMPLEXITY RESULTS
31
Given a game with a winning condition specified by an extendible condition type,
we can add vertices to the arena without significantly changing the size of the instance.
This enables us to assume that the arena has a very simple structure.
Theorem 2.59. Let A be an extendible regular condition type and G = (A, Ω) be a
Muller game with Ω ∈ A(A). Then there exists a Muller game (A0 , Ω0 ) with Ω0 ∈
A(A0 ), computable in time polynomial in ||G||, such that:
(i) A0 is a bipartite graph with E(A0 ) ⊆ (V0 (A0 ) × V1 (A0 )) ∪ (V1 (A0 ) × V0 (A0 )),
(ii) All vertices in V0 (A0 ) have out-degree at most 2, and
(iii) Player 0 wins G if, and only if, she wins G0 .
Proof. We construct A0 from A in a series of stages by adding vertices and adding
and replacing edges, so V (A) ⊆ V (A0 ). We observe that the resulting arena has
size polynomial in |A|, so it can be constructed in polynomial time. We then use
the definition of extendible condition type to obtain the winning condition Ω0 from
Ω. Since the size of A0 is polynomial in the size of A, we can compute Ω0 in time
polynomial in |Ω| + |A|. It is clear from the definition of extendible condition types
that in the resulting game Player 0 wins from vI (A) if, and only if, she wins from
vI (A0 ). Thus it remains to show the first two conditions may be met with at most a
polynomial increase in the size of the arena.
First we ensure all vertices in V0 (A0 ) have out-degree at most 2. If v ∈ V0 (A) has
out-degree m > 2, we replace the m outgoing edges from v with a binary branching
tree, rooted at v, with m leaves – the successors of v. We observe that this requires
adding at most m vertices and m edges. Each of the newly added vertices are added
to V1 (A). After repeating this for all vertices in V0 (A), the resulting arena A0 has at
most |V (A)| + |E(A)| vertices, and 2|E(A)| edges, and every vertex in V0 (A0 ) has
out-degree at most 2.
Now suppose all vertices in A have out-degree at most 2. For each edge e =
(u, v) ∈ E(A) such that u, v ∈ V0 (A) (u, v ∈ V1 (A)), add a vertex ve to V1 (A)
(V0 (A)) and replace the edge e with edges (u, ve ) and (ve , v). After repeating this for
all edges in E(A), the resulting arena A0 has at most |V (A)| + |E(A)| vertices, and
2|E(A)| edges, and E(A0 ) ⊆ V0 (A0 ) × V1 (A0 ) ∪ V1 (A0 ) × V0 (A0 ).
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2.3 Complexity results
In this section we consider the complexity of deciding whether Player 0 has a winning
strategy in a Muller game when the winning condition is specified using some of the
formalisms we have considered. We show that the problem of deciding Muller games in
which the winning condition is specified by a win-set condition is P SPACE -complete. It
follows from our results on translatability that the decision problems for Muller games
with winning condition specified by a Muller condition, Zielonka DAG or an EmersonLei condition are all also P SPACE-complete. We also show that the decision problems
for basis and superset games are co-NP-complete.
We first consider some upper bounds. A well-known result is that simple games
can be decided in linear time.
32
CHAPTER 2. INFINITE GAMES
Theorem 2.60. Let G = (A, F ) be a simple game. Whether Player 0 wins G can be
decided in time O(|E(A)|).
Proof. Suppose F = ∅, the case when F = P(V (A)) is dual. Let W ⊆ V1 (A) be the
set of vertices in V1 (A) with no outgoing edges. We observe that Player 0 wins from
vI (A) if, and only if, Player 0 can force the play to a vertex v ∈ W . Thus, Player 0
0
has a winning strategy if, and only if, vI (A) ∈ ForceA
(W ). The required complexity
bound then follows from Lemma 2.18.
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t
In [IK02], Ishihara and Khoussainov considered the following restriction on explicitly presented Muller games:
Definition 2.61 (Fully Separated game). Let G = (A, F ) be an explicitly presented
Muller game. We say G is fully separated if for each X ∈ F there exists vX ∈ X such
that vX ∈
/ Y for all Y ∈ F, Y 6= X.
Khoussainov showed that the winner of a fully separated game can be decided in
time O(|V (A)|2 |E(A)|). We now prove a generalization of this result by showing that
explicitly presented Muller games can be decided in polynomial time if the winning
condition is an anti-chain with respect to the subset relation.
Theorem 2.62. Let G = (A, F ) be an explicitly presented Muller game such that F
is an anti-chain, that is, X 6⊆ Y for all X, Y ∈ F. Whether Player 0 wins G can be
decided in time O(|F ||V (A)|2 |E(A)|).
Proof. Consider the algorithm A NTICHAIN(A, F ) in Algorithm 2.2. We show that it
is correct and returns in time O(|F ||V (A)|2 |E(A)|).
Algorithm 2.2 A NTICHAIN(A, F )
Returns: true if, and only if, Player 0 has a winning strategy from vI (A) in (A, F )
when F is an anti-chain.
for each X ∈ F do
let NX = {v : SPlayer 0 has a winning strategy from v in the game (A, {X})}
let N = ForceA0 ( X∈F NX )
if vI (A) ∈ N then
return true
else if N = ∅ then
return false
else
let F 0 = {X ∈ F : X ∩ N = ∅}
return A NTICHAIN(A \ N, F 0 )
We first show that A NTICHAIN(A, F ) returns true if, and only if, Player 0 has a
winning strategy in G = (A, F ). Let us suppose N has been computed as above. We
consider three cases:
(i) vI (A) ∈ N . From the definition of N , there exists v ∈ V (A) and X ∈ F
such that Player 0 can force the play to v from vI (A) and Player 0 has a winning
2.3. COMPLEXITY RESULTS
33
strategy from v which visits every vertex in X, and only vertices in X, infinitely
often. The winning strategy for Player 0 is then to force the play to v and play
this strategy. Since X ∈ F, this is a winning strategy.
(ii) N = ∅. In this case, for every X ∈ F, Player 1 has a strategy τX from every
vertex in A which can ensure either not all vertices of X are visited infinitely
often, or some vertices not in X are visited infinitely often. The strategy for
Player 1 on (A, F ) is as follows. Play anything until the play enters some X ∈ F,
then play the strategy τX until the play leaves X. Clearly if there is no X ∈ F
such that the play remains forever in X, Player 1 wins the play. So let us suppose
the play remains indefinitely in X for some X ∈ F. From the definition of τX ,
the set I of vertices visited infinitely often is properly contained in X. Since F is
an anti-chain, it follows that I ∈
/ F. Thus Player 1 wins the play.
(iii) N 6= ∅ and vI (A) ∈
/ N . In this case, Player 1 can force the play to remain in
A\ N and it follows from case (i) above that Player 0 has a winning strategy from
every vertex in N . Clearly, if Player 0 has a winning strategy in (A \ N, F 0 ) then
she has a winning strategy in the larger game: if Player 1 chooses to keep the play
in A\N then Player 0 can play her winning strategy on the subgame, otherwise if
Player 1 chooses to move to a vertex in N , Player 0 can play her winning strategy
from N . Conversely, if Player 1 has a winning strategy in (A \ N, F 0 ) then, as
he can force the play to remain in A \ N , he can play his winning strategy on the
subgame.
Thus, A NTICHAIN(A, F ) returns true if, and only if, Player 0 has a winning strategy
in G = (A, F ).
To show the algorithm returns in time O(|F ||V (A)|2 |E(A)|), we require the following result from [IK02]:
Lemma 2.63 ([IK02]). Let G = (A, F ) be an explicitly presented Muller game with
F = {X}. Whether Player 0 has a winning strategy from a vertex v ∈ V (A) can be
decided in time O(|V (A)||E(A)|).
It follows that at each stage of the recursion, it takes O(|F ||V (A)||E(A)|) time to
compute N . Furthermore, since |N | ≥ 1 whenever A NTICHAIN(A, F ) is recursively
called, it follows that the algorithm has recursion depth at most |V (A)|. Thus the
algorithm runs in time O(|F ||V (A)|2 |E(A)|) as required.
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2.3.1
P SPACE -completeness
As we saw in Theorem 2.15, McNaughton [McN93] presented an algorithm for deciding Muller games in space O(|V (A)|2 ). In fact, the games he considered were win-set
games. However, the algorithm is easily adapted to the case where the winning condition is presented explicitly, or as a Muller condition, a Zielonka DAG, an Emerson-Lei
condition, or a circuit condition without significant increase in the space requirements.
Thus, each of these classes of games is decidable in P SPACE .
We now show corresponding lower bounds. By the results of the previous section,
it suffices to establish the hardness result for the win-set condition type.
CHAPTER 2. INFINITE GAMES
34
x0
5
ϕ
}z
..
.
/ x0 ∧ xk−1 ∧ ¬xk
..
.
*
¬x0 ∧ xk−1 ∧ xk
7
¬x0
..
.
)
¬xk−1
5 xk−1 cF
O FF
x; O
FF xxx
xxFF
xx FFF
xx
+ ¬x
/ xk
k
Figure 2.3: Arena of GΦ for ϕ = (x0 ∧ xk−1 ∧ ¬xk ) ∨ . . . ∨ (¬x0 ∧ xk−1 ∧ xk )
Theorem 2.64. Deciding win-set games is P SPACE -complete.
Proof. By the above comments, we only need to show P SPACE-hardness. For this, we
reduce the problem of QSAT (satisfiability of a quantified boolean formula [QBF]) to
the problem of deciding the winner of a win-set game.
We assume, without loss of generality a QBF, Φ = Qk−1 xk−1 . . . ∀x1 ∃x0 ϕ is
given in which quantifiers are strictly alternating and ϕ is in disjunctive normal form
with 3 literals per clause. We then define a win-set game GΦ = (A, Ω), where Ω =
(W, W), as follows:
• V0 (A) = {ϕ} ∪ {x, ¬x : for all variables x},
• V1 (A) = {C0 , . . . , Cm−1 }, the set of clauses in ϕ,
• E(A) given by:
– (ϕ, Cj ) ∈ E(A) for 0 ≤ j < m;
– If Cj = (l0 ∧ l1 ∧ l2 ), then (Cj , l0 ), (Cj , l1 ), (Cj , l2 ) ∈ E(A);
– (xi , xi−1 ), (xi , ¬xi−1 ) ∈ E(A) for 0 < i < k;
– (¬xi , xi−1 ), (¬xi , ¬xi−1 ) ∈ E(A) for 0 < i < k; and
– (x0 , ϕ), (¬x0 , ϕ) ∈ E(A),
• vI (A) = ϕ,
• W = V0 (A) \ {ϕ}, and W is
W = Si , Si ∪ {xi }, Si ∪ {¬xi } : 0 ≤ i < k, i even
where S0 = ∅ and for i > 0, Si = {xj , ¬xj : 0 ≤ j < i}.
2.3. COMPLEXITY RESULTS
35
Figure 2.3 illustrates how the arena of GΦ would look if ϕ contained the clauses
(x0 ∧ xk−1 ∧ ¬xk ) and (¬x0 ∧ xk−1 ∧ xk ).
Note that as this is a win-set game, we are only interested in vertices of W that
are visited infinitely often. Observe that the winning condition ensures that Player 0
can win if, and only if, the minimum i such that at most one of xi and ¬xi is visited
infinitely often is even. The idea behind the strategy for Player 0 is to perpetually verify
ϕ. The choice of strategies by both players then dictates the choices of the truth values
for each of the variables, and the winning condition guarantees a winning strategy for
Player 0 if, and only if, Φ is true. To formally show that Player 0 has a winning strategy
if, and only if, Φ is true, we proceed by induction on k, the number of quantifiers of Φ.
Base case: k = 1 By the idempotence of ∧ and ∨ and assuming Φ is closed, Φ is
logically equivalent to one of the following forms.
• Φ = ∃x0 .x0 or ∃x0 .¬x0 . In this case the arena consists of four vertices, {ϕ, C0 , x0 , ¬x0 }.
Player 0 wins by always returning to ϕ from whichever of x0 and ¬x0 Player 1
is forced to play to, and Φ is clearly true.
• Φ = ∃x0 .(x0 ∨ ¬x0 ). Here Φ is also true. The arena consists of five vertices
{ϕ, C0 , C1 , x0 , ¬x0 } and Player 0 has the only choice (at ϕ and x0 ). A winning
strategy is to always play from ϕ to C0 , and to return immediately to ϕ from x0 .
• Φ = ∃x0 .(x0 ∧ ¬x0 ). Here Φ is false. The arena consists of four vertices
{ϕ, C0 , x0 , ¬x0 } and Player 1 can force the play to visit both x0 and ¬x0 infinitely often by alternately choosing each from C0 . Note that this strategy requires memory to remember which vertex was visited last time.
Note that if x0 does not appear in ϕ, we can add the clause (x0 ∧¬x0 ) without changing
the truth value of Φ.
Inductive case: The inductive hypothesis asserts that if Φ has k − 1 quantifiers and
is closed, then Player 0 has a winning strategy if, and only if, Φ is true. To show that
this implies the case for k quantifiers, we use the following lemma which shows how
subgames correspond to restricted subformulas. First we introduce some notation. If
x is free in ϕ and v is either true or false, we write ϕ[x 7→ v] to denote the formula
obtained by substituting v for x in ϕ and simplifying. Note that if ϕ[x 7→ true] simplifies to true then ϕ must have at least one clause containing the single literal x, and if it
simplifies to false, then all clauses contain ¬x. The crucial lemma can now be stated as
Lemma 2.65. If Φ = Qxϕ (Q ∈ {∃, ∀}) and ϕ[x 7→ true] does not simplify to true
or false, then Gϕ[x7→true] is isomorphic to the subgame of GΦ = (A, Ω) induced by the
1
set AvoidAvoid
0 (¬x) (x). Dually, if ϕ[x 7→ false] does not simplify to true or false, then
A
1
Gϕ[x7→false] is isomorphic to the subgame of GΦ induced by the set AvoidAvoid
0 (x) (¬x).
A
Proof. ϕ[x 7→ true] consists of the clauses of ϕ that do not contain ¬x, with all occurrences of x removed. The assumption that ϕ[x 7→ true] does not simplify to true or
false implies that there is at least one such clause. The arena for the game Gϕ[x7→true]
thus consists of vertices for ϕ[x 7→ true], the clauses, and the variables (and their negations) of ϕ, excluding x and ¬x. The edges are the same as those for GΦ restricted to
36
CHAPTER 2. INFINITE GAMES
1
this vertex set. We show that the subarena of GΦ induced by AvoidAvoid
(x) is iden0
A (¬x)
tical. As the winning condition only depends on vertices corresponding to variables, it
follows that the winning conditions are also identical.
0
In GΦ = (A, Ω), the set AvoidA
(¬x) consists of the vertices from which Player 0
can avoid ¬x. As Player 1 chooses the play from vertices corresponding to clauses, the
set of vertices from which Player 1 can reach ¬x is {¬x} ∪ {C : ¬x ∈ C}. As there
is at least one clause that does not contain ¬x, Player 0 can play to that clause to avoid
¬x from ϕ. The only other vertex from which it is possible to reach ¬x is x (as x is
the outermost variable in Φ), and from there Player 0 can play to either y (for the next
outermost variable y) or ϕ (if no such variable exists). Thus
0
AvoidA
(¬x) = V (A) \ {¬x} ∪ {C : ¬x ∈ C} .
0
Next we consider AvoidV1 0 (x) for V 0 = AvoidA
(¬x). As ϕ does not contain a clause
containing x by itself, Player 0 cannot force the play to x from ϕ, as Player 1 can
always choose to play to another literal. Furthermore, as x is the outermost variable
in Φ, the only edges to x are from vertices associated with clauses. Thus x is the only
vertex from which Player 0 can force the play to visit x, so
AvoidV1 0 (x) = V 0 \ {x}.
1
Thus AvoidAvoid
(x) = V (A) \ {x, ¬x} ∪ {C : ¬x ∈ C} , which is precisely the
0
A (¬x)
vertex set of Gϕ[x7→true] . The edges for both arenas are those of GΦ restricted to these
vertices, as are the winning conditions. Thus the two games are identical.
a
To complete the inductive step, we consider two cases.
• Φ = ∃xk−1 .ϕ. If Φ is true, then there is a truth value v such that ϕ[xk−1 7→ v]
is true. Assume that v = true, the case for v = false being similar. The winning
strategy for Player 0 is then to avoid ¬xk−1 and try to play to xk−1 , playing
through each vertex in Sk−1 when the latter vertex is reached. Note that to
play through each vertex in Sk−1 requires at least two visits to xk−1 – Player 0
must remember (the parity of) the number of times she has visited that vertex.
If ϕ[xk−1 7→ v] simplifies to true, then Player 0 can force the play to visit
xk−1 , by playing to the clause that only contains xk−1 . Otherwise Player 1
1
can play to avoid xk−1 , restricting the play to AvoidAvoid
(xk−1 ). From
0
A (¬xk−1 )
the above lemma, this subgame is equivalent to Gϕ[xk−1 7→true] , and from the
inductive hypothesis, Player 0 has a winning strategy on this game. Thus the
strategy of Player 0 is to play her winning strategy on the smaller game. If Φ is
false, then Player 1 plays a strategy similar to the strategy of Player 0 in the case
below.
• Φ = ∀xk−1 .ϕ. In this case, if Φ is true, then for both choices of truth value
v ∈ {true, false}, ϕ[xk−1 7→ v] is true. The winning strategy for Player 0 is
to alternately attempt to play to each of xk−1 and ¬xk−1 (and then through all
vertices in Sk−1 ), avoiding the other at the same time. If, at any point, Player 1
plays to avoid the vertex Player 0 is attempting to reach, Player 0 plays her
2.3. COMPLEXITY RESULTS
37
winning strategy on the reduced game (which exists from the lemma and the
inductive hypothesis). Again, if Φ is false, Player 1 plays a strategy similar to
the strategy of Player 0 in the previous case. Note that in this case Player 0
cannot force the play to visit both xk−1 and ¬xk−1 .
u
t
From our work on translatability in Section 2.2 and our observation regarding
the P SPACE solvability of these games, we obtain completeness results for Muller
games when the winning condition is presented as a Muller condition, Zielonka DAG,
Emerson-Lei condition or a circuit condition.
Corollary 2.66. The following problems are P SPACE -complete: Deciding Muller games
with winning condition specified by a Muller condition, deciding Zielonka DAG games,
deciding Emerson-Lei games, and deciding circuit games.
It can be verified that an explicit presentation of the winning condition constructed
in the proof of Theorem 2.64 would be exponentially larger than the presentation using
a win-set. Thus, the proof cannot be used to provide a P SPACE-hardness result for the
explicitly presented games. The exact complexity of deciding the winner of such games
remains open. Indeed, it is conceivable (though it appears unlikely) that the problem is
in P TIME.
Open problem 2.67. Determine the precise complexity of deciding explicitly presented
Muller games.
Bounded tree-width arenas
In Chapter 4 we present a graph parameter known as tree-width. Tree-width is a measure of how closely a graph resembles a tree. It has proved useful in the design of
algorithms as many problems that are intractable on general graphs are known to have
polynomial time solutions when restricted to graphs of bounded tree-width. In the context of Muller games, Obdrz̆álek [Obd03] exhibited a polynomial-time algorithm for
deciding the winner in parity games on arenas of bounded tree-width. We show that
this is not the case for Muller games (and neither, therefore, for Zielonka DAG games,
Emerson-Lei games, and circuit games). The proof of Theorem 2.64 can be modified
so that the arenas constructed all have tree-width two provided we allow ourselves to
specify the winning condition as a Muller condition rather than a win-set.
Theorem 2.68. Deciding Muller games specified by a Muller condition on arenas of
tree-width 2 is P SPACE -complete.
Proof. Membership of P SPACE follows from the fact that deciding general Muller
games specified by a Muller condition is in P SPACE .
The construction to show P SPACE -hardness is similar to that of Theorem 2.64. The
reduction is also from QSAT, and the proof that it is in fact a reduction is similar. Given
a QBF Φ = Qk−1 xk−1 . . . ∀x1 ∃x0 ϕ where ϕ is in DNF with three literals per clause,
the Muller game we construct is:
• V1 (A) = D where D is the set of clauses.
CHAPTER 2. INFINITE GAMES
38
GF
x0 ∧ xk−1 ∧ xk
66 HH
v:
66 HHH
vv
v
66 HHH
vv
v
66 HH$
vvv
66
..
ϕ
66 xk−1
.
66
66
66
xk
x0
/
/ ···
/ xk−1
ED
E
BCD
/
x0
/ ¬x0
/ ¬xk−1
/ ···
ED
BC
/ ¬x0
BC
Figure 2.4: Arena with bounded tree-width
• V0 (A) = {ϕ} ∪ D × {1, 2, 3} × {x, ¬x : x is a variable} .
• We have the following edges in E(A) for all c ∈ D:
– (ϕ, c),
– c, (c, n, l) if l is the n-th literal in c,
– (c, n, xi ), (c, n, xi−1 ) if the n-th literal of c is xi (i > 0)
– (c, n, x0 ), ϕ if the n-th literal of c is x0
– (c, n, xi ), (c, n, ¬xi ) for all i less than the index of the n-th literal of c
– (c, n, ¬xi ), (c, n, xi−1 ) for all i less than or equal to the index of the n-th
literal of c
– (c, n, ¬x0 ), ϕ for all n.
• C = {ϕ} ∪ {x, ¬x : x is a variable} is the set of colours,
• χ : V (A) → C defined as:
– χ(ϕ) = χ(c) = ϕ for all c ∈ D
– χ (c, n, l) = l.
• C = Si , Si ∪ {xi }, Si ∪ {¬xi } : 0 ≤ i < k, i even where S0 = {ϕ} and for
i > 0, Si = {ϕ} ∪ {xj , ¬xj : 0 ≤ j < i}.
Figure 2.4 illustrates how this arena differs from that of Theorem 2.64.
The resulting arena has tree-width 2, and the proof that Player 0 has a winning
strategy if, and only if, Φ is true is similar to that of Theorem 2.64.
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t
2.3. COMPLEXITY RESULTS
39
2.3.2 Complexity of union-closed games
We now turn our attention to Muller games where the winning condition F is a unionclosed set. Among games studied in the literature, Streett games and parity games are
examples of condition types that can only specify union-closed games. Union-closed
games were also studied as a class in [IK02]. One consideration that makes them an
interesting case to study is that they admit memoryless strategies for Player 1 [Kla94].
That is, on a game with a union-closed winning condition, if Player 1 has a winning
strategy then he has a strategy which is a function only of the current position. One
consequence of this fact is that, for explicitly presented union-closed games, the problem of deciding whether Player 0 wins such a game is in co-NP. This is because once
a memoryless strategy for Player 1 is fixed, the problem of deciding whether Player 0
wins against that fixed strategy is in P TIME. Indeed, it is a version of a simple game.
Thus, to decide whether Player 1 has a winning strategy we can nondeterministically
guess such a strategy and then verify that Player 0 cannot defeat it. Hence, determining whether Player 1 wins is in NP and therefore deciding whether Player 0 wins is in
co-NP. In this section, we aim to establish a corresponding lower bound for two condition types that can only represent union-closed games, namely the basis and superset
condition types.
We saw with Theorem 2.53 that we cannot use the known complexity bounds on
Streett games to easily establish similar bounds for basis games. Nevertheless, deciding
basis games is still in co-NP.
Proposition 2.69. Deciding basis games is in co-NP.
Proof. From the comments above, it suffices to show that if we fix a memoryless strategy for Player 1 then we can decide the resulting single player basis game in polynomial
time.
The algorithm is as follows. Let B be the basis for the winning condition. Initially
let B0 = B, and repeat the following:
S
1. Let Xi = B∈Bi B.
2. Partition Xi into strongly connected components (SCCs).
3. Remove any element of Bi which is not wholly contained in a SCC to obtain
Bi+1 ,
until Bi = Bi−1 , at which point, let X = Xi . This takes at most O |B|(|V (A)| +
|E(A)|) time using a standard SCC-partitioning algorithm. At this point, every SCC
of X is a union of basis elements – all x in X are members of basis elements, and any
basis elements not contained in any SCC of X is removed at step 3. Furthermore, any
strongly connected set of V (A) which is a union of basis elements is a subset (of an
SCC) of X, because the algorithm preserves such sets. Thus, Player 0 can win from
any node from which she can reach X (play to X and then visit every node within an
SCC of X forever); and Player 0 cannot win if she cannot reach X (there is no union
of basis elements for which Player 0 can visit every vertex infinitely often). Thus the
set of nodes from which Player 0 wins can be computed in O |B|(|V | + |E|) + |E|
time.
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CHAPTER 2. INFINITE GAMES
40
We now obtain the lower bounds we seek on superset games.
Theorem 2.70. Deciding superset games is co-NP-complete.
Proof. Membership of co-NP follows from Propositions 2.54 and 2.69. To show co-NPhardness, we use a reduction from validity of DNF formulas.
Given a formula ϕ(x0 , x1 , . . . , xk−1 ) in DNF, consider the superset game defined
as follows:
• for every variable xi we include three vertices, xi , ¬xi ∈ V0 (A) and x0i ∈
V1 (A);
• for each i we have the edges (x0i , xi ), (x0i , ¬xi ), (xi , x0i+1 ), (¬xi , x0i+1 ), where
addition is taken modulo k;
• vI (A) = x0 ; and
• the winning condition is specified by the set
M = {li ∈ V0 (A) : li is a literal of C} for every clause C of ϕ ,
As the superset condition is closed under union, if Player 1 has a winning strategy
he has a memoryless winning strategy. Note that any memoryless strategy for Player 1
effectively chooses a truth value for each variable. The set of vertices visited infinitely
often is a superset of an element of M if, and only if, the truth assignment chosen by
Player 1 makes one clause of ϕ (and hence ϕ) true. Thus Player 0 wins this game if,
and only if, there is no truth assignment which makes ϕ false.
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Corollary 2.71. Deciding basis games is co-NP-complete.
We note in conclusion that the exact complexity of deciding union-closed games
when they are explicitly presented remains an open problem. It is clearly in co-NP but
the above arguments do not establish lower bounds for it.
Open problem 2.72. Determine the precise complexity of deciding explicitly presented
union-closed games.
2.4 Infinite tree automata
One of the original motivations for studying Muller and related games was to establish
decidability results for problems such as non-emptiness and model checking for infinite
tree automata [McN66]. A reduction to non-emptiness of infinite tree automata is
used in some of the most effective algorithms for deciding satisfiability of formulas in
logics such as S2S, µ-calculus, CTL∗ , and other logics useful for reasoning about nonterminating, branching computation. Furthermore, determining if a structure satisfies
a formula in any of these logics reduces to determining if a certain automaton accepts
a particular tree. In this section we show that the non-emptiness and model-checking
problems (for regular trees) are P SPACE -complete for Muller automata. We first present
the definitions of infinite trees and infinite tree automata.
2.4. INFINITE TREE AUTOMATA
41
Definition 2.73 (Infinite tree). For k ∈ N, let [k] = {1, 2, . . . , k}. An infinite, k-ary
branching tree labelled by elements of Σ is a function t : [k]∗ → Σ. Nodes of an
infinite tree are elements of its domain, the root of an infinite tree is the empty string.
Definition 2.74 (Regular tree). A subtree of tree t rooted at u ∈ [k]∗ is the tree tu
defined as tu (v) = t(u · v) for all v ∈ [k]∗ . A tree t is regular if it has finitely many
distinct subtrees, or equivalently, if there are finitely many equivalence classes under
the equivalence relation
u ∼ v ⇐⇒ t(u · w) = t(v · w)
∀w ∈ [k]∗ .
Note that if a tree is regular it can be represented by a finite transition system, with
the equivalence classes of the above equivalence relation as states, the equivalence class
containing the root as the initial vertex, and k distinct transition relations.
Definition 2.75 (Infinite tree automaton). An infinite (Muller) (k-ary) tree automaton
is a tuple A = (Q, Σ, δ, q0 , F ) where
• Q is a finite set of states
• Σ is a finite alphabet
• δ ⊆ Q × Σ × Qk is a transition relation
• q0 is the initial state
• F ⊆ P(Q) is the acceptance condition.
Given an infinite, k-ary branching tree t labelled by elements of Σ, a run of A on t
is an infinite, k-ary branching tree r labelled by elements of Q satisfying the following
two conditions.
• The root of r is labelled by q0 (r() = q0 ).
• For all w ∈ [k]∗ , if r(w) = q, r(w · 1) = q1 , r(w · 2) = q2 , . . . , r(w · k) = qk ,
and t(w) = a, then (q, a, q1 , q2 , . . . , qk ) ∈ δ.
We say a run r is successful if for every (infinite) path, the set I of states visited
infinitely often is an element of F . We say A accepts t if there is a successful run of A
on t. Given an automaton A, the language of A is the set of trees
L(A) := {t : A accepts t}.
Two important decision problems in automata theory are non-emptiness and modelchecking.
N ON - EMPTINESS OF M ULLER TREE AUTOMATA
Instance: A Muller automaton A
Problem: Is L(A) 6= ∅?
CHAPTER 2. INFINITE GAMES
42
M ODEL - CHECKING FOR M ULLER TREE AUTOMATA
Instance: A Muller automaton A, and a regular infinite tree t
Problem: Is t ∈ L(A)?
The close connection between automata and games can be established by considering the game where the moves of Player 0 consist of choosing a transition in δ to make
from a current state, and the moves of Player 1 consist of choosing which branch of the
tree to descend. With this translation in mind, the non-emptiness problem
reduces to
the problem of finding the winner in the win-set game A, (W, W) with
• V0 (A) = W = Q,
• V1 (A) = Qk ,
• W = F,
• edges from V0 (A) to V1 (A) determined by δ: an edge from q to (q1 , q2 , . . . , qk )
if there is a ∈ Σ such that (q, a, q1 , . . . qk ) ∈ δ, and
• edges from V1 (A) to V0 (A) being projections: an edge from (q1 , . . . , qk ) to qi
for all i ∈ [k].
Clearly if Player 0 has a winning strategy in this game, it is possible to construct a tree
which the automaton accepts. Conversely, if Player 1 has a winning strategy, no such
tree exists.
By adapting the proof of Theorem 2.64 we are able to show that the non-emptiness
problem for Muller automata as well as the problem of determining whether a given
automaton accepts a given regular tree are both P SPACE -complete.
Theorem 2.76. The non-emptiness problem for Muller tree automata is P SPACE -complete.
Proof. Membership in P SPACE is established by the above polynomial time reduction
from the non-emptiness problem of Muller automata to win-set games. Here we show
P SPACE hardness through a reduction from QSAT (satisfiability of a quantified boolean
formula [QBF]).
Given a QBF Φ = Qk−1 xk−1 . . . ∀x1 ∃x0 ϕ, where ϕ is in disjunctive normal
form with 3 literals per clause, we construct the following Muller automaton AΦ =
(Q, Σ, qI , δ, F ) that accepts infinite ternary trees:
• Q = {qϕ } ∪ {qx , q¬x : for all variables x}
• Σ = {a} 2
• qI = qϕ
• δ ⊆ Q × Q3 given by:
– for each clause (l0 ∧ l1 ∧ l2 ) ∈ ϕ, (qϕ , ql0 , ql1 , ql2 ) ∈ δ;
2 as
Σ is a singleton, for ease of reading we omit a from the description of δ
2.4. INFINITE TREE AUTOMATA
43
– (qxi , qxi−1 , qxi−1 , qxi−1 ) ∈ δ for 0 < i < k;
– (q¬xi , qxi−1 , qxi−1 , qxi−1 ) ∈ δ for 0 < i < k;
– (qx0 , qϕ , qϕ , qϕ ) ∈ δ; and
– (q¬x0 , qϕ , qϕ , qϕ ) ∈ δ.
• F = Si , Si ∪ {qxi }, Si ∪ {q¬xi } : 0 ≤ i < k, i even where Si = {qϕ } ∪
{qxj , q¬xj : 0 ≤ j < i}.
Now by using the reduction to win-set games outlined above, asking if AΦ accepts any
tree is equivalent to asking if Player 0 has a winning strategy (from qϕ ) on the win-set
game used in Theorem 2.64.
u
t
The model checking problem also reduces to deciding which player wins an infinite
game. However, depending on how the tree is presented, the resulting arena may be of
infinite size. If the tree is presented as a finite transition system, a game with finite arena
can be constructed, and we can apply Theorem 2.76 to obtain the following corollary.
Corollary 2.77. Given a regular, infinite, k-ary branching tree t (represented as a
transition system) and a Muller automaton A = (Q, Σ, δ, qI , F ), asking if A accepts t
is P SPACE-complete.
Proof. P SPACE hardness follows from the proof of Theorem 2.76, as the automata
constructed there accept at most one tree – the ternary branching tree with all nodes
labelled by a.
To show that the problem is in P SPACE , we reduce it to the problem of deciding a
Muller game. Let (S, s0 , t1 , . . . , tk ) denote
the transition system representing the tree
t. The required Muller game, A, (χ, C) , is given by the following.
• V0 (A) = Q × S.
• V1 (A) = Q × S × Qk .
• There is an edge from (q, s) ∈ V0 (A) to (q, s, q1 , . . . qk ) ∈ V1 (A) whenever
(q, a, q1 , . . . , qk ) ∈ δ where a is the label of s.
• There is an edge from (q, s, q1 , . . . , qk ) ∈ V1 (A) to (qi , ti (s)) ∈ V0 (A) for
1 ≤ i ≤ k.
• vI (A) = (qI , s0 ),
• Q is the set of colours,
• χ : V (A) → Q is defined by taking the first component of the vertex.
• C = F.
It is clear from the definitions that Player 0 has a winning strategy from (qI , s0 ) in this
game if, and only if, A accepts t.
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44
CHAPTER 2. INFINITE GAMES
Chapter 3
Strategy Improvement for
Parity Games
In Chapter 2 we introduced parity games and briefly remarked on the significance of
determining the complexity of deciding them. One factor contributing to the importance of the analysis of parity games is that deciding the winner of a parity game
is polynomial-time equivalent to the model-checking problem of modal µ-calculus, a
highly expressive fragment of monadic second order logic [EJS01]. Indeed, the modal
µ-calculus is the bisimulation invariant fragment of monadic second order logic, and
therefore includes logics useful for verification such as the branching time temporal
logic CTL∗ [Dam94].
Another interesting aspect of parity games is that the complexity of deciding the
winner remains tantalizingly elusive. In Section 2.3 we observed that when we can restrict one player to memoryless strategies we can nondeterministically guess the strategy and if we can check in polynomial time if that strategy is winning, we have demonstrated an NP algorithm (if Player 0 has a memoryless winning strategy) or a co-NP
algorithm (if Player 1 has a memoryless strategy). So, from Theorem 2.42 we obtain
the following corollary:
Corollary 3.1. Deciding the winner of a parity game is in NP ∩ co-NP.
It is believed by some that parity games are decidable in polynomial time, however
the problem has so far resisted attempts to find tractable algorithms, giving us the
following well-researched open problem:
Open problem 3.2. Determine the exact complexity of deciding parity games.
In this chapter, we analyse one of the best candidates for a tractable algorithm for
parity games: the strategy improvement algorithm. In Chapters 6 and 7 we define a
large class (indeed, the largest class so far known) of graphs on which parity games
can be solved in polynomial time.
Currently, the√best known algorithm for deciding a parity game on general arenas
runs in time nO( n/ log n) where n is the number of vertices of the arena [JPZ06].
45
46
CHAPTER 3. STRATEGY IMPROVEMENT FOR PARITY GAMES
If p
the number of priorities, p, is small compared to the size of the arena, say p =
o( n/ log n), we can
slightly improve on this with an algorithm that runs in time
bp/2c n
where m is the number of edges of the arena [Jur00]. However,
O dm · bp/2c
in [VJ00a], Vöge and Jurdziński introduced a strategy improvement algorithm which
appears to do quite well in practice, even when the number ofQ
priorities is large.To date,
the best known upper bound for its running time is O mn v∈V0 (A) dout (v) , which
is in general exponential in the number of vertices. However, no family of examples
has yet been found that runs in worse than linear time. In this chapter we analyse the
structure of this algorithm and use combinatorial results to improve the known upper
and lower bounds. The analysis we use is primarily taken from [VJ00b].
3.1 The strategy improvement algorithm
The idea behind the strategy improvement algorithm is to define a measure dependent
on the strategy of Player 0. Then, starting with an arbitrary strategy for Player 0, to
make local adjustments based on this measure to obtain a new strategy which is in
some sense improved. This process is then repeated until no further improvements
can be made. At this point, with a judicious choice of measure, the strategy is the
optimal play for Player 0, and the winning sets for each player can easily be computed.
This procedure is readily extended to any strategy that requires finite memory, so from
Theorem 2.14 we see that it can be used for games other than parity games. However,
with parity games we can restrict ourselves to memoryless strategies and then at each
stage both the measure and the local improvements can be efficiently computed.
In order to fully describe the algorithm, we need to introduce some concepts. Using
the notation of Chapter 2, let us fix a parity game G = (A, χ) where χ : V (A) → P.
For convenience we assume no vertex in A has out-degree 0. For the remainder of this
chapter, we assume all strategies are positional.
To be able to evaluate strategies, we first identify the characteristics of a play which
are important. A play profile is a triple (l, P, e) where l ∈ P, P ⊆ P and e ∈ ω. Given
an infinite play π = v1 v2 · · · in G, we associate with π a play profile, Θ(π) := (l, P, e),
as follows. We define l to be the maximum priority occurring infinitely often in χ(π),
so the parity of l determines the winner of the play. We define P to be the set of
priorities greater than l that occur in χ(π), and e to be the minimal index such that
χ(ve ) = l and χ(ve0 ) ≤ l for all e0 ≥ e. A valuation is a mapping from each vertex
v ∈ V (A) to a play profile of an infinite play from v.
We next define an ordering that compares play profiles by how beneficial they are
to each player. We begin by defining a useful linear order on the set of priorities. The
reward order, v, is defined as follows: for i, j ∈ P, i v j if either
(i) i is odd and j is even, or
(ii) i and j are even and i ≤ j, or
(iii) i and j are odd and i ≥ j.
Intuitively, i v j if j is “better” for Player 0 than i. We extend v to play profiles by
defining (l, P, e) @ (m, Q, f ) if either
3.1. THE STRATEGY IMPROVEMENT ALGORITHM
47
(i) l @ m; or
(ii) l = m and max≤ (P 4 Q) is odd and in P , or even and in Q; or
(iii) l = m, P = Q, and either l is odd and e < f , or l is even and e > f .
The measure we use to implement the strategy improvement algorithm is a valuation that gives the v-minimal play profile amongst all plays consistent with the current
strategy for Player 0. More precisely, let σ be a strategy for Player 0, and for v ∈ V (A)
let Playsσ (v) be the set of all infinite plays starting from v consistent with σ. We define
the valuation ϕσ by:
ϕσ (v) := min v {Θ(π) : π ∈ Playsσ (v)}.
The next proposition, taken from [VJ00b], helps give an intuitive understanding of
ϕσ . Given a strategy σ for Player 0 and a strategy τ for Player 1, we observe there is
precisely one infinite play πστ (v) consistent with σ and τ from each vertex v ∈ V (A).
We write Θστ for the valuation defined by:
Θστ (v) := Θ πστ (v) .
If we further extend v to a partial order on valuations, E, in a pointwise manner then
Proposition 5.1 of [VJ00b] can be stated as:
Proposition 3.3. The set {Θστ : τ is a strategy for Player 1} has a E-minimal element
and it is equal to ϕσ .
Intuitively, this means that ϕσ is equivalent to the valuation defined by σ and the best
counter-strategy for Player 1 against σ. Consequently, ϕσ can be efficiently computed
by fixing the strategy of Player 0 and considering the strategies of Player 1 in the
resulting single player game.
After computing ϕσ , the algorithm makes local improvements to the strategy σ by
switching (if necessary) σ(v) to the successor of v with the v-maximal ϕσ value. The
resulting strategy σ 0 is improved in the sense that ϕσ E ϕσ0 . This is then repeated
until no further improvements can be made. At this point the strategy σ is optimal for
Player 0, that is, Player 0 can win from a vertex v ∈ V (A) against any strategy for
Player 1 if, and only if, she can win playing σ from v against any strategy. We can then
compute the winning sets by fixing Player 0’s strategy and finding the winning sets for
Player 1 in the single player game. Algorithm 3.1 provides a detailed description of the
critical part of the strategy improvement algorithm.
As an example, let us consider the parity game pictured in Figure 3.1. Let σ be the
strategy for Player 0 defined by σ(a) = a0 , σ(b) = b0 and σ(c) = c1 . We will compute
ϕσ for the vertices a0 , b0 and b1 . Against σ, Player 1 has a choice of strategies at a0 :
either he can play to c, resulting in an infinite play with maximum priority 4, or he can
play to a, resulting in an infinite play with maximum priority 1. As 1 @ 4, the latter is
the v-minimal choice and so ϕσ (a0 ) = (1, ∅, 0). At b0 , Player 1’s choice appears to
depend on the strategy at a0 : if he plays to a and the strategy at a0 is to play to a then
the resulting play has maximum priority 1, otherwise if the strategy at a0 is to play to c
the resulting play has maximum priority 4. However 3 @ 1, so the v-minimal play in
either case is going to be to play to b, resulting in ϕσ (b0 ) = (3, ∅, 0). The valuation at
48
CHAPTER 3. STRATEGY IMPROVEMENT FOR PARITY GAMES
Algorithm 3.1 Strategy optimization
Returns: An optimal strategy for Player 0
select a strategy σ for Player 0 at random
repeat
let σ = σ 0
{Store current strategy}
Compute ϕσ
for each v ∈ V0 do
{Improve σ locally according to ϕσ }
select w such that (v, w) ∈ E(A) and
ϕσ (w) = maxv {ϕσ (v 0 ) : (v, v 0 ) ∈ E(A)}
if ϕσ σ(v) @ ϕσ (w) then
let σ 0 (v) = w
until σ = σ 0
return σ
/ a :1
0
c 0 : 6 iT
b0 : 3 Tj T
TTTT TTTT
jjj
 ju jjjjj
T)
T*
tjjjj
a : 0 TTT
b : 0 TTTT
4
j
j5 c : 0
j
j
j
TTT)
_
TTT*
jjjj
jjjj
a1 : 2
b1 : 5
c1 : 4 o
Figure 3.1: A parity game
3.2. A COMBINATORIAL PERSPECTIVE
σ
000
001
010
011
100
101
110
111
i
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
ϕσ (ai )
ϕσ (bi )
ϕσ (ci )
(3, {6}, 4)
(3, ∅, 0)
(3, {6}, 2)
(3, ∅, 2)
(3, {5, 6}, 4)
(3, {4, 6}, 6)
(1, ∅, 0)
(3, ∅, 0)
(3, {6}, 2)
(3, ∅, 2)
(1, {4, 5}, 4)
(1, {4}, 2)
(1, ∅, 0)
(1, {3}, 2)
(6, ∅, 0)
(6, ∅, 4)
(6, ∅, 2)
(1, {4}, 2)
(1, ∅, 0)
(1, {3, 4, 5}, 6) (1, {4, 5, 6}, 6)
(1, {2, 4, 5}, 6) (1, {4, 5}, 4)
(1, {4}, 2)
(3, ∅, 4)
(3, ∅, 0)
(3, {6}, 2)
(3, ∅, 2)
(3, {5, 6}, 4)
(3, {4}, 4)
(3, ∅, 4)
(3, ∅, 0)
(3, {6}, 2)
(3, ∅, 2)
(3, {4, 5}, 6)
(3, {4}, 4)
(6, ∅, 6)
(6, ∅, 6)
(6, ∅, 0)
(6, ∅, 4)
(6, ∅, 2)
(6, ∅, 6)
(5, ∅, 4)
(5, ∅, 2)
(5, {6}, 2)
(5, ∅, 2)
(5, ∅, 0)
(5, ∅, 4)
49
σ0
VID
011
011
011
010
110
100
010
001
010
110
000
101
110
000
000
111
Table 3.1: Table of valuations, next strategy and improvement vectors for all strategies
b1 is only dependent on the choice of strategy at a0 , so ϕσ (b1 ) = (1, {4, 5}, 4). Turning
to the subsequent, improved strategy σ 0 , we have (3, ∅, 0) @ (1, {4, 5}, 4). Therefore,
switching σ at b will be an improvement for Player 0, and hence σ 0 (b) = b1 .
Using ijk as shorthand for the strategy which maps a to ai , b to bj , and c to ck , the
full table of relevant valuations and subsequent strategies for each strategy is presented
in Table 3.1. Also included in this table is the vector of improving directions (VID),
indicating which elements of σ had improvements. Not only does this help identify
110 as the optimal strategy, but it is worth observing that each entry in the VID column
is unique. As we see in the next section, this is not a coincidence.
3.2 A combinatorial perspective
In this section we show how the strategy improvement algorithm can be viewed as an
optimization problem on a well-studied combinatorial structure. We will introduce the
concepts of acyclic unique sink oriented hypercubes and the bottom-antipodal sinkfinding algorithm and we will prove the following result:
Theorem 3.4. The strategy improvement algorithm is a bottom-antipodal sink-finding
algorithm on an acyclic unique sink orientation of the strategy hypercube.
Although this result appears in [BSV03], we present an alternative proof that utilises
results from [VJ00b].
50
CHAPTER 3. STRATEGY IMPROVEMENT FOR PARITY GAMES
•
/•
•
/•
O
•
/•
O
•O
/•
•
/•
•
/•
•o
•
•o
•
Figure 3.2: AUSOs of the 2-cube (l) and the orientations which are not AUSOs (r)
First we recall some definitions relating to hypercubes. A d-dimensional hypercube
is an undirected graph Hd such that V (Hd ) = {0, 1}d, and there is an edge between
(a1 , . . . , ad ) and (b1 , . . . , bd ) if for some i ≤ d, ai 6= bi and aj = bj for all j 6= i.
We call ai the i-th component of a vertex (a1 , . . . , ad ) in a d-dimensional hypercube.
A subcube is a subgraph induced by a set of vertices which agree on some set of components. We observe that a subcube of a d-dimensional hypercube is a d0 -dimensional
hypercube for some d0 ≤ d, and we can specify a subcube by a single vertex together
with a set of adjacent edges. Given a set I ⊆ {1, . . . , d} of natural numbers and a
vertex v = (a1 , . . . , ad ) of a d-dimensional hypercube, we denote by SwitchI (v) the
vertex v 0 = (b1 , . . . , bd ) obtained by switching the components in I of v. That is,
bi = ai if, and only if, i ∈
/ I. Given a vertex v in a d-dimensional hypercube, the
vertex antipodal to v is the vertex Switch{1,...,d} (v).
Given a parity game (A, χ), we assume that every vertex in V0 (A) has out-degree
two. From Theorem 2.59, we can always transform a parity game into one for which
every vertex in V0 (A) has out-degree at most two. We can assume there are no vertices
of out-degree 0, as we can use force-sets to determine if either player can force the play
to one of these vertices. We can also change any vertex in V0 (A) with out-degree 1 to
be a vertex in V1 (A) as this does not affect the outcome of the game. As this can all
be done in polynomial time, this assumption is not too restrictive. If we fix an order
on V0 (A) = {v1 , . . . , vd }, and write vi0 and vi1 for the two successors of vi ∈ V0 (A),
then each vector (b1 , . . . , bd ) ∈ {0, 1}n defines a strategy for Player 0 by mapping vi
to vibi , and conversely each strategy defines a unique vector. Therefore, the space of
all Player 0’s strategies is equivalent to vertex set of the d-dimensional hypercube. For
convenience, we will simply refer to the strategy space as the strategy hypercube. We
now introduce some additional concepts to help establish Theorem 3.4.
An orientation of a d-dimensional hypercube is a directed graph with a d-dimensional
hypercube as an underlying undirected graph and at most one edge between any pair
of vertices. We say an orientation is an acyclic unique sink orientation (AUSO) if it is
acyclic and every subgraph induced by a subcube has a unique sink (or, equivalently,
a unique source). Figure 3.2 shows the two AUSOs for the 2-cube (left), together with
the two orientations of the 2-cube which are not AUSOs (right).
Acyclic unique sink orientations of hypercubes are very important combinatorial
structures, particularly as a generalization of linear programming optimization problems. For example, a pseudo-boolean function (PBF) is a function from a hypercube
to R, and a common optimization problem is to find the vertex which attains the maximum (or minimum) value of a PBF. In [HSLdW88], a hierarchy of classes of PBFs
was introduced, and one of these classes was the completely unimodal pseudo-boolean
functions: functions such that every subcube has a unique local minimum. Clearly, a
3.2. A COMBINATORIAL PERSPECTIVE
51
completely unimodal PBF induces an AUSO, and conversely any function to R which
respects an AUSO will be completely unimodal.
One useful concept associated with AUSOs is the vector of improving directions.
Let VID : {0, 1}n → {0, 1}n be the function that maps each vertex of a hypercube with
an AUSO to the vector which indicates which edges are outgoing from that vertex. That
is, if there is an edge from v to v 0 where v and v 0 differ in the i-th component, then the
i-th component of VID(v) is 1 and the i-th component of VID(v 0 ) is 0.
An important class of problems for AUSOs and similar structures are polynomial
local search problems (PLS). These are optimization problems where the cost of a
solution and “neighbouring” solutions can be efficiently computed, with the overall
goal being to find a locally optimal solution – one which is better than all its neighbours.
For example, if computing the directions of edges incident with a vertex can be done in
polynomial time, then finding the unique global sink of an acyclic unique sink oriented
hypercube is a problem in PLS. Clearly, given a hypercube we could iterate through
all vertices to find the sink, but as is usually the case for interesting problems in PLS,
iterating through all possible solutions is considered infeasible. For the sink-finding
problem a more interesting question is: can we find the global sink in time polynomial
in the dimension of the hypercube? In fact, for acyclic unique sink oriented hypercubes,
this is an important open problem.
Open problem 3.5. Given an n-dimensional hypercube with an AUSO, is there a polynomial p such that the global sink can be found with at most p(n) vertex queries?
One reason for the importance of this question is that there are interesting structural
results for AUSOs that suggest this question can be answered in the affirmative. Firstly,
an n-dimensional hypercube with an AUSO satisfies the Hirsch conjecture [WH88],
which means that from each vertex there is a directed path of length at most n to the
global sink. Secondly, we have the following observation from Williamson Hoke [WH88]
which shows that the vector of improving direction takes a very special form:
Theorem 3.6 ([WH88]). VID is a bijection.
However, despite these results, an efficient sink-finding algorithm on hypercubes with
AUSOs remains elusive.
The connection between AUSOs and the strategy improvement algorithm is summarized in the following theorem:
Theorem 3.7. The valuation ϕσ induces an AUSO on the strategy hypercube.
In order to prove this, we must first indicate how ϕσ induces an orientation. Let <
be any linear ordering on the set of Player 0’s strategies. We extend E to a partial order
on strategies by saying σ C σ 0 if either
(i) ϕσ C ϕσ0 , or
(ii) ϕσ = ϕσ0 and σ < σ 0 .
This gives us an orientation on the strategy hypercube, as we see with the following
result:
52
CHAPTER 3. STRATEGY IMPROVEMENT FOR PARITY GAMES
Lemma 3.8. Let σ and σ 0 be strategies for Player 0 such that σ(v) = σ 0 (v) for all but
one v ∈ V0 (A). Then either σ C σ 0 , or σ 0 C σ.
The proof of this result follows directly from the following two results from [VJ00b].
Lemma 5.7 of [VJ00b]. Let I ⊆ {1, . . . , d} be a set ofnatural numbers, and let σ
be a strategy for Player 0. If, for each i ∈ I, ϕσ σ(vi ) @ ϕσ (vi0 ) where vi0 is the
successor of vi not equal to σ(vi ), then σ E SwitchI (σ).
Claim 7.2 of [VJ00b]. Let I ⊆ {1, . . . , d} be a set
of natural numbers, and let σ be a
strategy for Player 0. If, for each i ∈ I, ϕσ σ(vi ) [email protected] ϕσ (vi0 ) where vi0 is the successor
of vi not equal to σ(vi ), then SwitchI (σ) E σ.
The orientation is then obtained by adding an edge from σ to σ 0 if σ(v) = σ 0 (v) for
all but one v ∈ V0 (A) and σ C σ 0 . We now need to show that this orientation is an
AUSO. To do this, we use the fact that the strategy improvement algorithm terminates.
Theorem 3.1 of [VJ00b]. The strategy improvement algorithm correctly computes the
winner of a parity game.
Since E is a partial order it is clear that this orientation is acyclic. In order to show
that it is an AUSO, we use the following result about unique sink orientations.
Proposition 3.9 ([WH88]). A hypercube orientation is a unique sink orientation if,
and only if, every 2-dimensional subcube has a unique sink.
Next we observe that every subcube of the strategy hypercube induces a subgame of
the original parity game: by definition, there is a set V ⊆ V0 (A) on which all strategies
of the subcube agree. The induced subgame is obtained by fixing Player 0’s choices
on V to agree with all the strategies of the subcube. Furthermore, in these subgames
ϕσ takes the same values as in the original parity game. Thus the resulting strategy
hypercube of the subgame is a subcube of the strategy hypercube of the original game.
Therefore, if any 2-dimensional subcube of the strategy hypercube does not have a
unique sink, we can produce a parity game with a 2-dimensional strategy hypercube
with the same orientation. The only acyclic orientation of a 2-cube without a unique
sink is one with antipodal sinks and sources (see Figure 3.2). In Lemma 3.10 we
describe how the strategy improvement algorithm works on an oriented hypercube,
and from this we see that if the algorithm begins at a source of this 2-dimensional
hypercube, then the subsequent strategy will always be the other source. Thus, on this
orientation, the algorithm never terminates. Since Theorem 3.1 of [VJ00b] ensures that
the strategy improvement algorithm always terminates, every 2-dimensional subcube
has a unique sink, and we have therefore shown that the orientation defined by C is an
AUSO. This completes the proof of Theorem 3.7.
Returning to the example parity game from the previous section, we can read the
orientation of the strategy hypercube directly from Table 3.1. For example, consider
the strategy σ = {001}. Since ϕσ (a1 ) @ ϕσ (a0 ), it follows that 101 E 001, thus
there is an edge from 101 to 001. Figure 3.3 shows the resulting oriented strategy
hypercube.
Having established that the set of strategies for Player 0 forms a hypercube oriented by C, we can investigate how the strategy improvement algorithm operates on
this cube. From Algorithm 3.1, we see that a strategy σ switches at each point where
3.3. IMPROVING THE KNOWN COMPLEXITY BOUNDS
53
110 o
111
? O






100 o
101
o
010
011
?
?




/
000
001
Figure 3.3: Oriented strategy hypercube for the parity game in Figure 3.1
ϕσ σ(v) is not v-maximal. If this is adjusted so that when there is a choice of
strategies with v-maximal ϕσ values, we choose the <-largest strategy, then from
Lemma 3.8 we see that we are switching σ at the vertices corresponding to the outgoing edges in the strategy hypercube. That is,
Lemma 3.10. Let σ be a strategy for Player 0 and Cσ be the subcube of the oriented
strategy hypercube defined by σ and the outgoing edges from σ. Then the subsequent
strategy σ 0 in the strategy improvement algorithm is the vertex antipodal to σ on Cσ .
This is a well-known sink-finding procedure for AUSO hypercubes called B OTTOM [SS05], described in Algorithm 3.2. It is clear that on an AUSO hyper-
ANTIPODAL
Algorithm 3.2 B OTTOM - ANTIPODAL
Returns: Global sink of an AUSO hypercube
select a vertex v at random
repeat
Compute VID(v)
let v = v ⊕ VID(v)
until VID(v) = 0
return v
{XOR v and VID(v)}
cube, B OTTOM - ANTIPODAL terminates with the global sink: at each stage we are
jumping from the unique source of the subcube defined by the improving directions to
some other vertex in that subcube, so we are always reducing the minimal distance to
the global sink. Combining Lemma 3.10 with Theorem 3.7 gives us the main result:
Theorem 3.4. The strategy improvement algorithm is a bottom-antipodal sink-finding
algorithm on an acyclic unique sink orientation of the strategy hypercube.
3.3 Improving the known complexity bounds
Q
The upper bound of O mn v∈V0 dout (v) for the running time of the strategy improvement algorithm
Q arises from theobservations that it takes O(mn) time to compute
ϕσ and there are
v∈V0 (A) dout (v) different strategies for Player 0 [VJ00a]. The results of Section 3.2 enable us to improve the trivial upper bound obtained by naı̈vely
54
CHAPTER 3. STRATEGY IMPROVEMENT FOR PARITY GAMES
running through all possible strategies. Mansour and Singh [MS99] showed that a
d
B OTTOM - ANTIPODAL sink-finding algorithm will visit at most O 2d vertices of a
d-dimensional hypercube. However, we can improve this upper bound further by using
results from combinatorics. Instead of using the B OTTOM - ANTIPODAL algorithm, we
can use other sink-finding algorithms such as the F IBONACCI S EE - SAW of Szabó and
Welzl [SW01], described in Algorithm 3.3. This algorithm utilises structural results
of AUSOs such as Theorem 3.6 and has the best-known running time upper bound,
O(1.61d ), amongst sink-finding algorithms.
Algorithm 3.3 F IBONACCI S EE - SAW
Returns: Global sink of an AUSO hypercube
select a vertex m at random
let w be the vertex antipodal to m
let Cm = {m} and Cw = {w}
{Antipodal i-dimensional subcubes}
for i = 0 to n do
Compute VID(m) = (m0 , m1 , . . .) and VID(w) = (w0 , w1 , . . .)
let d = min{j : mj 6= wj }
0
let Cm
be the i-dimensional subcube parallel to Cm in direction d from m
0
let Cw be the i-dimensional subcube parallel to Cw in direction d from w
if md = 1 then {m is the minimal vertex of an (i + 1)-dimensional subcube}
0
Compute w = F IBONACCI S EE - SAW(Cw
)
else
{w is the minimal vertex of an (i + 1)-dimensional subcube}
0
Compute m = F IBONACCI S EE - SAW(Cm
)
0
0
let Cm = Cm ∪ Cm and Cw = Cw ∪ Cw
return m
These results give us the following improved upper bounds for the strategy improvement algorithm:
Proposition 3.11. Assuming each vertex in V0 (A) has out-degree two:
(i) The strategy improvement algorithm runs in time O(mn · 2n0 /n0 ).
(ii) The Fibonacci strategy improvement algorithm runs in time O(mn · 1.61n0 ).
Where m = |E(A)|, n = |V (A)| and n0 = |V0 (A)|.
Turning to lower bounds, natural questions to consider are completeness results. In
particular, is strategy improvement or finding the sink of an AUSO hypercube PLScomplete? Björklund et al. [BSV03] show that this is not the case.
Theorem 3.12 ([BSV03]). The problem of finding optimal strategies in parity games
is not PLS-complete with respect to tight PLS-reductions.
Because PLS-complete problems have exponentially long improvement paths [Yan97],
the fact that strategy improvement is not PLS-complete gives further support to the
hypothesis that it may only require polynomially many iterations.
3.3. IMPROVING THE KNOWN COMPLEXITY BOUNDS
55
However, we can also ask if there are examples of parity games which require an
exponential number of strategies to be considered by the strategy improvement algorithm. As a first step towards this, Schurr and Szabó [SS05] generated a family of
oriented hypercubes for which B OTTOM - ANTIPODAL visits 2d/2 vertices. It remains
an open problem whether there is a family of parity games with these hypercubes as
their strategy hypercubes. In fact, this can be generalized to a more interesting open
problem:
Open problem 3.13. Given a hypercube with an AUSO, can a parity game be constructed in polynomial time with that hypercube as its strategy hypercube?
A positive answer to this question would not only give an exponential worst case for the
strategy improvement algorithm, but it would also relate Open Problems 3.2 and 3.5: a
polynomial time algorithm for finding the sink on an AUSO would give a polynomial
time algorithm for solving parity games and vice versa. On the other hand a negative answer to this question would give a smaller class of AUSOs for which finding a
polynomial time sink-finding procedure is an interesting and important problem.
This leads to another interesting question: Can we classify the AUSO hypercubes
that correspond to parity games? As we mentioned previously, Hammer et al. [HSLdW88]
introduced a hierarchy of pseudo-boolean functions including completely unimodal
functions. It seems plausible that the class of PBFs corresponding to parity games
might lie within one of the more restrictive families they considered. For example,
viewing a d-dimensional hypercube Hd as a polytope in Rd , a PBF ϕ on Hd is linearly separable if for all r ∈ R there exists a hyperplane separating the vertices v with
ϕ(v) ≥ r from the vertices v 0 with ϕ(v 0 ) < r. It is easily seen that a divide-andconquer algorithm can find the sink of a linearly separable hypercube in time linear in
the dimension, so if the hypercube orientations associated with parity games are linearly separable then the strategy improvement algorithm would run in polynomial time.
However, as the next result shows, the hierarchy of [HSLdW88] is not fine enough to
separate parity games and completely unimodal functions. We say a pseudo-boolean
function f : {0, 1}n → R is pseudomodular if for all v, w ∈ {0, 1}n:
(i) min{f (v), f (w)} ≤ max{f (v ∧ w), f (v ∨ w)}, and
(ii) min{f (v ∧ w), f (v ∨ w)} ≤ max{f (v), f (w)}.
In [HSLdW88], the class of pseudomodular functions was the least restrictive class of
PBFs included in completely unimodal functions. However,
Proposition 3.14. There exists a parity game with an oriented strategy hypercube that
cannot be induced by a pseudomodular function.
Proof. Consider the parity game from Figure 3.1. Its oriented strategy hypercube can
be seen in Figure 3.3. We see that
111 C 000 C 001 C 110.
Now taking v = 001 and w = 110 we see that there is no function f : {0, 1}3 → R
that can simultaneously respect C and satisfy both pseudomodular axioms above. u
t
56
CHAPTER 3. STRATEGY IMPROVEMENT FOR PARITY GAMES
This result is not surprising, there is no obvious reason why the joins and meets
of strategies should satisfy the pseudomodular conditions. However, it does imply for
instance that there are strategy hypercubes which are not linearly separable.
Chapter 4
Complexity measures for
digraphs
In the last few chapters we examined the computational complexity of some graphbased games. We saw how the winning condition influences the difficulty of the problem of finding a winner of such games. We now turn our attention to the other aspect
of such games, the arena. The aim of the next few chapters is to investigate measures
of graph complexity, in particular measures for directed graphs. As we will see, such
metrics give insight into the structure theory of graphs and help identify those characteristics that act as a barrier to finding efficient solutions of various important problems
(for example, finding the winner of a parity game, or finding a Hamiltonian path). Consistent with the overall theme of this dissertation, the complexity measures we define
will be based on games.
So what makes a good complexity measure? First we have to consider what it is
we are aiming to measure. This of course depends largely on the application one has in
mind. For instance, a group theorist may be interested in graph automorphisms and so
a useful measure might reflect the size of the automorphism group. A topologist might
be interested in a measure that indicates how many edges must cross in a drawing of
the graph on a surface, or how many paths there are between any pair of vertices. We
are interested in algorithmic aspects, so a practical measure might indicate the difference between tractable and intractable instances of many NP-complete problems. A
good measure of complexity may even encompass more than one such aim. So one
desirable property is soundness: the measure can be defined in equivalent ways for
different applications. Another desirable property is robustness: the measure should be
“well-behaved”. For example, if we simplify the graph, then the measure should not
increase. Again, the concept of simplification is dependent on the application. For the
group theorist, a simple graph is one in which all vertices have similar structure, for
example, a clique. For the topologist a simple graph might be an acyclic graph. From
the algorithmic perspective, simplifying would include operations that likely reduce
the complexity of many problems, for instance taking subgraphs. In this case simple
graphs would be a class on which many NP-complete problems have polynomial time
57
58
CHAPTER 4. COMPLEXITY MEASURES FOR DIGRAPHS
solutions – again, acyclic graphs are a good example. Dually, if we complicate the
graph the measure should not decrease, and if this complication is in some way uniform, we would expect the measure to increase uniformly. One final desirable feature,
particularly for algorithmic purposes, is that the measure should somehow encompass
large classes of graphs. For example, acyclicity is a sound and robust measure, but
it only takes two values, a graph is either acyclic or it is not. So although acyclicity
provides a boundary between tractable and intractable instances of many NP-complete
problems, we cannot use it to find larger classes of graphs which may admit efficient
solutions. This suggests that a generalization of acyclicity, perhaps indicating how
acyclic a graph is, would be an ideal candidate for a good complexity measure. This is
precisely the type of measure we consider in this and the following chapters.
In this chapter we introduce an important and well-known measure for undirected
graphs called tree-width. We show how it matches the criteria outlined above, and
we discuss the problem with its extension to directed graphs, providing motivation for
subsequent chapters.
4.1 Tree-width
Tree-width can be seen as a measure of graph complexity for both topological and
algorithmic purposes. That it serves both purposes is not surprising as it is often the
complexity of the structure of the graph that makes problems difficult to solve; many
NP-complete problems can be solved in polynomial time on the topologically simple
class of acyclic graphs. As the name suggests, the tree-width of a graph indicates how
close that graph is to being a tree. For example, trees have tree-width 1, simple cycles
have tree-width 2, and highly connected graphs such as cliques have tree-width one
less than the number of vertices in the graph.
Although Robertson and Seymour coined the name tree-width [RS84], the parameter had been around for many years prior to this, testament to the importance of
tree-width as a measure of graph complexity. Rose and Tarjan [RT75] considered a
symbolic approach to Gaussian elimination on matrices which amounts to vertex elimination on graphs. They introduced several parameters which reflect how “difficult”
it is to perform a sequence of eliminations: for example the width of an elimination
reflects the maximum number of operations required at any stage of the elimination.
The minimum width over all vertex eliminations is a graph measure equivalent to treewidth. Halin [Hal76] considered S-functions: mappings from graphs to integers satisfying certain formal conditions, a class of functions which includes graph parameters
such as the chromatic number, the vertex-connectivity and the homomorphism-degree.
Halin showed that there is a maximal S-function under the natural point-wise partial
ordering of S-functions, and this function turns out to be the tree-width of the graph.
Arnborg [Arn85] was one of the first to show the algorithmic importance of tree-width,
by finding efficient solutions to many NP-complete problems on partial k-trees, a characterization of the class of graphs with tree-width bounded by k. We will revisit some
of these alternative characterizations of tree-width in Chapter 7.
To formally define tree-width, we must first introduce the notion of a tree decomposition. A tree decomposition of a graph G is an arrangement of subgraphs of G in a
4.1. TREE-WIDTH
59
tree-like manner so that all paths in the graph respect this arrangement. More precisely,
Definition 4.1 (Tree decompositions and tree-width). Let G be an undirected graph. A
tree decomposition of G is a pair (T , X ) where T is a tree and X = (Xt )t∈V (T ) is a
family of subsets of V (G) such that:
S
(T1) X is a cover of V (G), that is, X∈X X = V (G),
(T2) For each vertex v ∈ V (G) the subgraph of T induced by the set {t : v ∈ Xt } is
a connected subtree, and
(T3) For each edge {u, v} ∈ E(G) there exists t ∈ V (T ) such that {u, v} ⊆ Xt .
The width of a decomposition (T , X ) is max{|Xt | : t ∈ V (T )} − 1. The tree-width
of a graph G, Tree-width(G) is the minimum width over all tree decompositions of G.
To see how this definition corresponds with our informal description above, let G
be an undirected graph and (T , X ) be a pair such that T is a tree and X = (Xt )t∈V (T )
is a cover of V (G). For an arc1 e = {s, t} ∈ E(T ), we observe that the removal of e
from T gives twoS
subtrees: one, Ts , containing
the node s, the other, Tt containing the
S
node t. Let Vs = t0 ∈Ts Xt0 and Vt = t0 ∈Tt Xt0 . We define the following condition:
(T4) For each arc {s, t} ∈ E(T ), every path from Vs to Vt contains at least one vertex
in Xs ∩ Xt .
Condition (T4) can be used as an alternative to conditions (T2) and (T3) as we see in
the following lemma.
Lemma 4.2. Let G be an undirected graph, and (T , X ) a pair such that T is a tree
and X = (Xt )t∈V (T ) is a cover of V (G). Then (T4) holds if, and only if, both (T2)
and (T3) hold.
Proof. Suppose (T4) holds. For each vertex v ∈ V (G), let T [v] be the subgraph of T
induced by the set {t ∈ V (T ) : v ∈ Xt }. Suppose T [v] is not connected. Let C1 and
C2 be two distinct components of T [v]. Since T is a tree, there is a unique path in T
from C1 to C2 . Let (s, s0 ) be the first arc in that path. Since C1 and C2 are distinct
components, we have s ∈ C1 and s0 ∈
/ V (T [v]), so v ∈ Xs ⊆ Vs , but v ∈
/ Xs0 , so
v∈
/ Xs ∩ Xs0 . However, C2 ⊆ Ts0 , so v ∈ Vs0 . As the path (of length 0) from v to
itself does not go through Xs ∩ Xs0 we have a contradiction. Thus (T2) holds. Now
let e = {u, v} be an edge of G and suppose T [u] and T [v] have no nodes in common.
Let (s, s0 ) be the first arc in the unique path from T [u] to T [v] in T . We observe that
/ Ts and v ∈ Ts0 . But then no vertex on the (length 1) path from u
u ∈ Ts , u ∈
/ Ts0 , v ∈
to v along e is contained in Xs ∩ Xs0 , a contradiction. Therefore, (T3) holds.
Now suppose (T2) and (T3) hold. Let {s, s0 } be an arc of T . Let (v1 , . . . , vn )
be a path from v1 ∈ Vs to vn ∈ Vs0 . We show that there must be some i such that
vi ∈ Xs ∩ Xs0 . If vi ∈ Vs ∩ Vs0 for any i, 1 ≤ i ≤ n, then it follows from (T2) that
vi ∈ Xs ∩ Xs0 and we are done. So assume that there is no i such that vi ∈ Vs ∩ Vs0 .
1 To assist with descriptions, we use the terms nodes and arcs when referring to T , and the terms vertices
and edges for G.
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CHAPTER 4. COMPLEXITY MEASURES FOR DIGRAPHS
Since v1 ∈ Vs and vn ∈ Vs0 , it follows that there is some j, 1 < j ≤ n such that vi ∈ Vs
for all 1 ≤ i < j, and vj ∈ Vs0 . But there is an edge from vj−1 to vj so from (T3) there
exists t ∈ V (T ) such that {vj−1 , vj } ⊆ Xt . Now V (Ts ) ∪ V (Ts0 ) = V (T ), so either
t ∈ V (Ts ) or t ∈ V (Ts0 ). In the first case it follows that vj ∈ Vs , and in the second it
u
t
follows that vj−1 ∈ Vs0 , both of which are contradictions. Therefore (T4) holds.
Path-width
Path-width, also introduced by Robertson and Seymour [RS83], is a measure of complexity for undirected graphs closely related to tree-width. Just as tree-width indicates
how close a graph is to being a tree, path-width indicates how close a graph is to being
a path. Indeed, a path decomposition is a tree decomposition in which the underlying
tree is a path. More precisely,
Definition 4.3 (Path decomposition and path-width). Let G be an undirected graph. A
path decomposition of G is a sequence X1 , . . . , Xn of subsets of V (G) such that:
Sn
(P1) i=1 Xi = V (G),
(P2) If i ≤ j ≤ k then Xi ∩ Xk ⊆ Xj , and
(P3) For each e = {u, v} ∈ E(G), there exists i ≤ n such that {u, v} ⊆ Xi .
The width of a path decomposition, X1 , . . . , Xn , is max{|Xi | : 1 ≤ i ≤ k} − 1. The
path-width of G is the smallest width of any path decomposition of G.
It is worth observing that if X1 , . . . , Xn is a path decomposition of a graph G, then
so is Xn , . . . , X1 . Thus a path decomposition is not completely dependent on the linear
order imposed by the fact that it is a sequence.
Because a path decomposition is also a tree decomposition, path-width is a weaker
notion of graph complexity than tree-width. That is, if a graph has path-width k, then
the graph has tree-width ≤ k. The difference between the two can be arbitrarily large:
the class of trees has tree-width 1, but unbounded path-width. However, as argued
in [DK05], path-width can be seen as a first approximation of tree-width, and many
interesting structural results can be established with the measure. For example, we
have the following result of Bienstock, Robertson, Seymour and Thomas:
Theorem 4.4 ([BRST91]). For every forest T , every graph of path-width ≥ |V (T )|−1
has a minor isomorphic to T .
4.1.1 Structural importance of tree-width
Lemma 4.2 gives us a good insight into what graph properties tree-width measures.
If we take the given definition of a tree decomposition, we see that tree-width is essentially a measure indicating how much structure we need to ignore before the graph
becomes acyclic. In this way, tree-width measures the cyclicity of a graph. On the
other hand, if we define tree decompositions using (T1) and (T4) we see that tree-width
measures how well separate parts of the graph are linked. In other words, tree-width
also measures the connectedness of a graph. Lemma 4.2 asserts that on undirected
4.1. TREE-WIDTH
61
graphs cyclicity and connectedness generalize to the same measure. As we will see,
this distinction is important, because on directed graphs cyclicity and connectedness
are significantly different, giving us a variety of complexity measures to consider.
In Chapter 1, we indicated that the concept of “graph structure” that we are interested in investigating is algorithmically motivated. As we have suggested, cyclicity
and connectedness are important algorithmic structural properties, so this suggests that
tree-width is a useful measure for graph structure.
An important relation for the theory of graph structure that we are investigating is
the minor relation. Intuitively the minor relation relates two graphs if one is structurally
“more complex” than the other. We formally define the concept in Chapter 8. It is
not surprising that tree-width and the minor relation are closely connected. Indeed,
tree-width was an important tool in the proof by Robertson and Seymour [RS04] of the
Graph Minor Theorem (see Theorem 8.42), described by Diestel as “among the deepest
results mathematics has to offer” [Die05]. In addition many other structural measures
have been shown to be intimately related to tree-width. For instance a feedback vertex
set is a set of vertices whose removal result in an acyclic graph. It is easy to show that
if a graph has a feedback vertex set of size k, then it has tree-width at most k + 1. Two
other important structural measures are havens and brambles.
Definition 4.5 (Haven). Let G be an undirected graph and k ∈ N. A haven of order k
in G is a function β : [V (G)]<k → P(V (G)) such that for all X ⊆ V (G) with |X| < k:
(H1) β(X) is a non-empty connected component of G \ X, and
(H2) If Y ⊆ X, then β(Y ) ⊇ β(X).
Definition 4.6 (Bramble). Let G be an undirected graph. A bramble in G is a set B of
connected subsets of V (G) such that for all pairs B, B 0 ∈ B either B ∩ B 0 6= ∅, or
there exists {u, v} ∈ E(G) with u ∈ B and v ∈ B 0 . The width of a bramble B is the
minimum size of a set which has a non-empty intersection with every element of B.
Seymour and Thomas [ST93] demonstrated the relation between havens, brambles
and tree-width with the following theorem:
Theorem 4.7 ([ST93]). Let G be an undirected graph. The following are equivalent:
1. G has tree-width ≥ k − 1
2. G has a haven of order k.
3. G has a bramble of width k.
This theorem asserts that the smallest width of all tree decompositions is always
equal to the largest width of all brambles. Since the width of tree decompositions is a
maximizing measure and the width of brambles is a minimizing measure, Theorem 4.7
is a minimax theorem. We explore this aspect of tree-width further in Chapter 8.
The importance of tree-width as a measure of structural complexity suggests that
tree-width is robust under various structural transformations, particularly those, such
as taking subgraphs, which may affect the complexity of problems. Indeed, this can
be verified by examining the definition of tree decompositions, but is perhaps best
illustrated by Theorem 5.37, which we present in the next chapter.
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4.1.2 Algorithmic importance of tree-width
The nature of tree decompositions further supports the algorithmic significance of treewidth, as the structure of a decomposition lends itself well to dynamic programming
techniques [Bod88]. When we restrict to a class of graphs of bounded tree-width, we
bound the size of the tree decompositions and many algorithms based on dynamic programming will run in polynomial time. Thus restricting to classes of graphs of bounded
tree-width can provide large classes of tractable instances for many NP-complete problems. This was best illustrated by Arnborg and Proskurowski [AP89], when they provided efficient algorithms for many well-known NP-complete problems on graphs of
bounded tree-width. This was further extended by Courcelle’s elegant characterization
of a large class of problems which can be efficiently solved with dynamic programming:
Theorem 4.8 ([Cou90]). Any problem which can be formulated in Monadic Second
Order logic can be solved in linear time on any class of graphs of bounded tree-width.
Of course the applicability of these results depends largely on the complexity of the
following decision problem:
T REE - WIDTH
Instance: An undirected graph G, and a natural number k
Problem: Is the tree-width of G at most k?
While this problem is NP-complete [ACP87], for a fixed value k determining if a
graph has tree-width k and indeed, computing a tree decomposition of width k if one
exists, can be performed in linear time [Bod96]. This means that finding the tree-width
of a graph is fixed parameter tractable, and so it is not surprising that tree-width has
also played a major role in advancing the field of parameterized complexity.
As we mentioned earlier many important graph parameters are closely related to
tree-width, so a common technique for finding fixed parameter tractable algorithms for
parameterized problems is to use tree-width to separate instances into those which can
be trivially solved and those which can be solved using bounded tree-width techniques.
For example, consider the parameterized problem of finding a feedback vertex set of
size k. We can use the fixed parameter tractable algorithm for computing tree-width
to compute a tree decomposition of width k + 1. If no such decomposition exists then
there cannot be a feedback vertex set of size k. Otherwise, since the feedback vertex
set problem can be formulated in MSO, Courcelle’s theorem implies there exists an
algorithm to solve the problem in linear time, giving us a fixed parameter tractable
algorithm for finding a feedback vertex set of size k.
4.1.3 Extending tree-width to other structures
The above discussion indicates that tree-width is a practical, sound and robust complexity measure for undirected graphs. We now consider other structures such as directed
graphs or hypergraphs. One key to the success of tree-width is that tree decompositions
are readily extendable to arbitrary relational structures. If, in Definition 4.1, we replace
“vertices” with “elements of the universe”, and condition (T3) with:
4.1. TREE-WIDTH
63
(T30 ) For each relation R and each tuple (a1 , a2 , . . .) in the interpretation of R there
exists t ∈ V (T ) such that {a1 , a2 , . . .} ⊆ Xt ,
then we obtain a definition of tree-width for general relational structures. Consequently, we can benefit from the algorithmic advantages of tree-width, such as a structure well-suited to dynamic programming, and obtain large classes of tractable instances of problems outside graph problems. But how good is tree-width as a measure
of complexity on these structures? It is easy to see that the tree-width of a structure is
precisely the tree-width of the Gaifman graph of that structure: the graph with vertex
set equal to the universe of the structure and an edge between any two elements that
occur in a tuple of a relation. The main drawback of this approach is that by considering the Gaifman graph, we lose information about the structure, and in some cases this
information loss may be crucial. For example, the Gaifman graph of a directed graph
is the undirected graph obtained by ignoring the orientation of the edges, so the treewidth of a directed graph is the tree-width of the underlying undirected graph. This
means that directed acyclic graphs (DAGs) can have arbitrary tree-width as any graph
can be the underlying graph of a DAG. However, many interesting problems based on
directed graphs are greatly simplified when restricted to DAGs, so we would expect
DAGs to have low complexity. This suggests that tree-width is not a good complexity
measure of directed graphs, especially for algorithmic purposes.
This leads to the following research problem, the investigation of which forms the
core of the remaining chapters.
Research aim. Find a complexity measure for directed graphs which generalizes treewidth.
Before we give an overview of the current status of this problem, we discuss what
exactly “generalizes tree-width” entails. First, we are interested in measures which
generalize tree-width as a measure. This has two aspects. As tree-width is defined for
directed graphs, we are not interested in measures that may be “worse than” tree-width.
In other words, we are searching for measures that are bounded above by tree-width.
On the other hand, we can view undirected graphs as directed graphs by interpreting an
undirected edge as a pair of anti-parallel edges – recall the definition of bidirection in
Section 1.1.2. So we can look for a measure which matches tree-width on undirected
graphs by using this transformation to directed graphs.
The second property of tree-width we are interested in generalizing is the structural
aspect. Many structural properties of graphs have natural extensions to directed graphs,
for example acyclicity or connectivity. A good generalization of tree-width to directed
graphs would reflect the behaviour of tree-width with regard to these properties. In
particular we expect structurally simple directed graphs such as DAGs and directed
cycles to have low complexity, but structurally complex directed graphs such as cliques
to have high complexity, just as trees and cycles have small tree-width and cliques have
large tree-width. Similarly, we expect that a reasonable measure would be robust under
the structural relations for directed graphs we considered in Section 1.1.2. For example,
we expect that the measure would not increase under the taking of subgraphs, or that
it would be possible to compute the measure on a graph from its strongly or weakly
connected components, or more generally from a pair of subgraphs which comprise a
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CHAPTER 4. COMPLEXITY MEASURES FOR DIGRAPHS
directed union. This last property was considered in [JRST01] as an important property
for the generalization of tree-width to directed graphs.
Finally, we are also interested in generalizing tree-width in the algorithmic sense.
We are particularly interested in being able to find efficient algorithms for interesting
problems on directed graphs of bounded complexity. Having some sort of decomposition which generalizes tree decompositions might be one way to achieve this.
4.2 Directed tree-width
In [JRST01], Johnson, Robertson, Seymour and Thomas introduced an extension of
tree-width to directed graphs known as directed tree-width. Informally, directed treewidth is based on a decomposition, known as an arboreal decomposition, which is
defined by generalizing Condition (T4). Formally, to define directed tree-width, we
require the following definition:
Definition 4.9 (Z-normal). Given two disjoint subsets Z and S of vertices of a digraph
G, we say S is Z-normal if for every directed path, v1 · · · vn , in G such that v1 , vn ∈ S,
either vi ∈ S for all 1 ≤ i ≤ n, or there exists j ≤ n such that vj ∈ Z.
Also, given a directed tree T with edges oriented away from a unique vertex r ∈
V (T ) (called the root), we write t > e for t ∈ V (T ) and e ∈ E(T ) if e occurs on
the unique directed path from r to t, and e ∼ t if e is incident with t. The following
concepts were introduced in [JRST01].
Definition 4.10 (Arboreal decompositions [JRST02]). An arboreal decomposition of
a digraph G is a tuple (T , B, W) where T is a directed tree with a unique root, and
B = (Bt )t∈V (T ) and W = (We )e∈E(T ) are families of subsets of V (G) that satisfy:
(R1) B is a partition of V (G) into non-empty sets, and
S
(R2) If e ∈ E(T ), then B≥e := {Bt |t > e} is We -normal.
The width of an arboreal
decomposition (T , B, W) is the minimum k such that for all
S
t ∈ V (T ), |Bt ∪ e∼t We | ≤ k + 1. The directed tree-width of a digraph G, dtw(G),
is the minimal width of all its arboreal decompositions.
It follows from this definition that directed tree-width does generalize tree-width as
a measure in the sense described above.
Towards showing that directed tree-width is also a structural generalization, Johnson et al. considered the natural generalization of havens (using strongly connected
components rather than connected components) and proved the following analogue of
Theorem 4.7:
Theorem 4.11 ([JRST01]). Let G be a directed graph.
1. If G has a haven of order k then G has directed tree-width ≥ k − 1.
2. If G has no haven of order k then G has directed tree-width ≤ 3k − 2.
4.2. DIRECTED TREE-WIDTH
65
Johnson et al. conjectured that the bound in the second item could be reduced to
≤ k − 1, showing an equivalence between havens and directed tree-width. However
Adler [Adl05] has shown that this is not the case. Safari [Saf05] showed that natural
generalization of brambles (using strongly connected sets rather than connected sets),
can also be related to havens and directed tree-width.
Theorem 4.12 ([Saf05]). For a directed graph G let H(G) be the largest order of a
haven in G, and B(G) the largest width of any bramble in G. Then
H(G) ≤ 2B(G) ≤ 2H(G),
and there exist graphs for which equality holds in either inequality.
Johnson et al. also demonstrated the algorithmic potential of directed tree-width,
firstly by providing a general algorithm scheme for finding efficient algorithms on digraphs of bounded directed tree-width, and secondly by using this scheme to produce
an algorithm which solves the following problem in polynomial time on graphs of
bounded directed tree-width:
k-D ISJOINT PATHS
Instance: A directed graph G, and a set of k pairs of (not necessarily disjoint) vertices {(s1 , t1 ), . . . (sk , tk )}
Problem: Are there k vertex disjoint paths P1 , . . . , Pk in G such
that for each i, Pi is a path from si to ti ?
A corollary of this result is that many other important NP-complete problems, such
as the Hamiltonian path and cycle problems, can be solved efficiently on graphs of
bounded directed tree-width.
Theorem 4.13 ([JRST01]). The following problems can be solved in polynomial time
on any class of directed graphs with bounded directed tree-width: Hamiltonian cycle,
Hamiltonian path, k-Disjoint paths, Hamiltonian path with prescribed endpoints, Even
cycle through a given vertex.
In terms of parameterized complexity, directed tree-width is also quite useful. Although there is no known algorithm for computing the exact directed tree-width of a
graph apart from a brute-force search, generalizing the approach used to compute treewidth in fixed parameter linear time gives us a fixed parameter tractable algorithm for
computing an approximation of directed tree-width. This means that we can use directed tree-width in a similar role as tree-width for finding fixed parameter tractable
algorithms for problems on directed graphs.
Johnson et al. conclude their paper by observing that several other more natural
extensions of tree decompositions to directed graphs are not appropriate as they are not
robust under simple graph operations. They highlight that one of the major problems
with defining a notion of tree-width for directed graphs is that on directed graphs many
other structural measures are not as closely linked as they are in the undirected case, as
we saw in Theorem 4.12.
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CHAPTER 4. COMPLEXITY MEASURES FOR DIGRAPHS
4.3 Beyond directed tree-width
So with a seemingly appropriate complexity measure defined, why is the generalization of tree-width to directed graphs still an interesting research problem? The answer
is that directed tree-width does not seem to complete the whole picture. For a start,
unlike with tree-width the definition is awkward, as is the given algorithm scheme,
and it is difficult to gain an intuitive understanding. The structure of arboreal decompositions is not as flexible as tree decompositions, which means we cannot provide
alternative forms of the decomposition which may be useful algorithmically (see, for
example, Theorem 6.28). This makes it challenging to develop algorithms outside of
those provided in [JRST01], suggesting directed tree-width is not as practical as it first
appears.
In addition, contrary to the claims made in [JRST01], directed tree-width is not robust under some very simple graph operations. Adler [Adl05] has shown that directed
tree-width may increase under the taking of butterfly minors (see Definition 8.28), and
it appears that this can be extended to showing that directed tree-width may increase
under the taking of subgraphs. However, it follows from Theorem 4.11, that this increase can only be by a constant factor, as havens are robust under these operations.
While this means that algorithmically directed tree-width is still a useful measure of
complexity, it lessens the importance of directed tree-width as a structural measure.
This was further shown by Adler, with the following result which shows that havens
are distinct from directed tree-width.
Theorem 4.14 ([Adl05]). There exists a directed graph G with no haven of order 4 and
directed tree-width 4.
This implies that we cannot reduce the bound in the second part of Theorem 4.11
to obtain an equivalence between havens and directed tree-width.
Nevertheless, in the next chapter we show that Theorem 4.11 implies that directed
tree-width at least approximates a good complexity measure for directed graphs. But
the picture is still not complete. The problem is that on directed graphs there is a difference between connectivity and reachability – if there is a path from u to v it does
not necessarily follow that u and v are in the same strongly connected component,
and similarly, if u and v are in the same weakly connected component, there may
not be a path from u to v. The tree-width of a directed graph can be seen as a measure of its weak connectivity, as tree-width is a connectivity measure that, on directed
graphs, ignores edge direction. Likewise, the definitions of directed tree-width and its
alternative characterizations suggest that directed tree-width is a measure of the strong
connectivity of a graph. So the question can be asked, “What, if anything, measures
the reachability, or directed connectivity, of a directed graph?” In Chapters 6 and 7
we address this question, introducing two distinct, but closely related measures which
seem to indicate directed connectivity. As strong connectedness implies reachability,
and reachability implies weak connectedness, it is not surprising that these measures
lie between tree-width and directed tree-width. We argue that as these measures are
closer to tree-width than directed tree-width is, they are more practical as a complexity
measure for directed graphs. In Chapter 8 we consider the structural implications of
4.3. BEYOND DIRECTED TREE-WIDTH
67
the question, endeavouring to find generalizations of havens, brambles and minors that
correspond to our measures.
An interesting follow-up question is “Should a good complexity measure for directed graphs be invariant under edge reversal?” As many important structural features
such as cycles or strongly connected sets are preserved under reversing edges, it would
seem that a good structural measure would be invariant under this operation. However,
from an algorithmic point of view edge direction is much more critical. Consider the
problem of trying to find a path between two vertices when it is not easy to compute
the edge relation, but it is relatively easy to compute the successors of a vertex. Such a
problem might arise for instance if we were considering the computations of a Turing
machine. On a tree where all edges are oriented away from a single vertex, finding
such a path could involve a lot of back-tracking, but with all edges oriented towards a
single vertex, the problem becomes significantly easier. Unlike directed tree-width, the
measures we introduce in Chapters 6 and 7 are not invariant under the edge reversal operation, providing further evidence that they are more suitable extensions of tree-width
from a practical point of view.
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Chapter 5
Graph searching games
With a view to finding good complexity measures for directed graphs, we now turn our
attention to a means of developing robust measures of graph complexity. We introduce
a game played between two players, one controlling a fugitive located on the graph,
and the other controlling a set of searchers whose purpose is to locate the fugitive.
Such games are useful for describing problems such as trying to locate a virus in a
network, or locate someone in a cave system. They can also be used to define measures
of graph complexity: we obtain various complexity measures by considering variants
of the game and the resources required by the searchers to locate the fugitive. Indeed,
the tree-width of a graph can be characterized by the minimum number of searchers
required to locate the fugitive in some of the variants we consider.
We first define a very general form of the game which encompasses many games
considered in the literature, for example [ST93, KP86, BG04, DKT97, FFN05, GLS01,
GM06]. This enables us to define some important concepts we use throughout the next
few chapters: plays, searches, strategies and monotonicity. After demonstrating how
this game includes other games considered in the literature, we introduce a general
framework for developing measures of graph complexity. In Section 5.4, we show
how these measures are robust under some basic graph operations such as taking subgraphs. Finally, we conclude the chapter by considering the complexity of the problem
of determining these graph parameters.
5.1 Definitions
The definitions we present in this chapter are applicable to both directed and undirected
graphs, though it is often necessary to assume we are working within only one of these
classes. Thus we use the term graph to refer to a structure with a single, binary edge
relation which may or may not be symmetric.
We recall from Definition 2.7, the definition of a simple game. The game we are
interested in is a simple game played on an arena defined by the graph to be searched.
That is,
69
CHAPTER 5. GRAPH SEARCHING GAMES
70
Definition 5.1 (Graph searching game). A graph searching game type is a function Γ
which maps a graph G to a triple (Ls , Lf , A) where Ls and Lf are sets of subsets of
Elts(G) and A is an arena which satisfy:
• ∅ ∈ Ls ,
• ∅∈
/ Lf , and Lf has a unique ⊆-maximal element Rmax ,
• V0 (A) ⊆ Ls × Lf consists of pairs (X, R) where X ∩ R = ∅,
• V1 (A) ⊆ Ls ×Ls ×Lf consists of triples of the form (X, X 0 , R) where X ∩R =
∅,
• vI (A) = (∅, Rmax ),
• If (X, R), (X 0 , X 00 , R0 ) ∈ E(A) then X = X 0 and R = R0 ,
• If (X, X 0 , R), (X 00 , R0 ) ∈ E(A) then X 0 = X 00 and for all r0 ∈ R0 there
is r ∈ R such that r and r0 are in the same (weakly) connected component of
G \ (X ∩ X 0 ), and
• If S ⊆ R, then for all S 0 such that (X, X 0 , S), (X 0 , S 0 ) ∈ E(A), there exists
R0 ⊇ S 0 such that (X, X 0 , R), (X 0 , R0 ) ∈ E(A).
Given a graph searching game type Γ, and a graph G, with Γ(G) = (Ls , Lf , A)
the graph searching game on G (defined by Γ(G)) is the simple game GΓG := (A, F ),
where F = ∅, so Player 1 wins all infinite plays. In a graph searching game elements
of V0 (A) are called positions (of the game), elements of V1 (A) are called intermediate
positions, and we call Player 0 the searchers and Player 1 the fugitive.
Intuitively, the game works as follows. A graph searching game on G is a game
played by a number of co-operating searchers against an omniscient fugitive. All entities occupy elements of G, however, while the locations of the searchers are known to
everyone, the location of the fugitive is not necessarily known, so the fugitive “occupies” a set of potential locations. When the game is at the position (X, R), X ∈ Ls
represents the location of the searchers, and R ∈ Lf represents the set of potential
fugitive locations. The initial position, (∅, Rmax ), thus indicates that at the beginning
there are no searchers on G and the fugitive may be anywhere on Rmax . The searchers
and fugitive move around G, but, as indicated by the edge relation of the arena, only
the fugitive is necessarily constrained by the topology of G.
From position (X, R), the searchers, if possible, choose a new set of locations X 0 .
If this is not possible then the fugitive has escaped and he wins. Otherwise, the game
proceeds to the intermediate position (X, X 0 , R). For ease of later descriptions, we
say the searchers on X \ X 0 have been removed while the searchers on X ∩ X 0 remain
stationary and the searchers on X 0 \ X will be placed after the fugitive has completed
his move.
The fugitive responds to the move of the searchers at each of his potential locations,
but he is not permitted to pass through any stationary searchers. However, he is omniscient and is aware of the impending occupation of X 0 \ X by the searchers that will be
5.1. DEFINITIONS
71
placed, and can modify his response accordingly. The final condition in the definition
of the arena of a graph searching game asserts that the responses of the fugitive at each
of his potential locations are somewhat independent: if the set of potential locations
is increased, then so are the sets of his potential responses. Some information about
the response of the fugitive may be available to the searchers, resulting in a (visible)
choice for the fugitive about the next set, R0 , of his potential locations. If he has no
such choice and no possible location to move to (R0 = ∅), then he has been captured
and the searchers win. Otherwise, the game proceeds to the position (X 0 , R0 ). This
whole process is represented in the graph searching game by moving the token from
(X, R) to the vertex (X, X 0 , R), and then to (X 0 , R0 ). If the fugitive can avoid capture
forever, then again he has escaped and he wins.
From this we can see that an arena of a graph searching game on G can be described
by defining the set of positions and a set of legal transitions between positions, essentially “ignoring” non-terminal intermediate positions. It follows that all plays ending
with a move from the fugitive can be fully described as a sequence of positions:
(X0 , R0 )(X1 , R1 ) · · · (Xn , Rn )
where (X0 , R0 ) = (∅, Rmax ) and for 0 ≤ i < n and for all r0 ∈ Ri+1 there is r ∈ Ri
such that r and r0 are in the same connected component of G \ (Xi ∩ Xi+1 ). We extend
this to include plays that are winning for the searchers by using Rn = ∅ to indicate that
the play ended at (Xn−1 , Xn , Rn−1 ). This motivates the following definition:
Definition 5.2 (Search). Let GΓG be a graph searching game on G defined by (Ls , Lf , A),
and let (X1 , R1 ) ∈ V0 (A). A proper search from (X1 , R1 ) in GΓG is a (possibly infinite)
sequence, (X1 , R1 )(X2 , R2 ) · · · , such that for all i ≥ 1:
• (Xi , Ri ) ∈ V0 (A),
• (Xi , Ri ), (Xi , Xi+1 , Ri ) ∈ E(A), and
• (Xi , Xi+1 , Ri ), (Xi+1 , Ri+1 ) ∈ E(A).
A complete search from (X1 , R1 ) in GΓG is a finite sequence (X1 , R1 ) · · · (Xn , Rn )
such that
• (X1 , R1 ) · · · (Xn−1 , Rn−1 ) is a proper search from (X1 , R1 ) in GΓG ,
• (Xn−1 , Rn−1 ), (Xn−1 , Xn , Rn−1 ) ∈ E(A),
• (Xn−1 , Xn , Rn−1 ) ∈ V1 (A) has no outgoing edges, and
• Rn = ∅.
A search in GΓG is a sequence which is either a proper or a complete search. A search
π can be extended to a search π 0 , if π is a prefix of π 0 . A search from vI (A) is winning
for the searchers if it can be extended to a complete search, otherwise it is winning for
the fugitive.
CHAPTER 5. GRAPH SEARCHING GAMES
72
In the sequel we will generally adopt this representation of plays as we are primarily
concerned with the game from the perspective of the searchers.
Variants of graph searching games are obtained by restricting the moves available
to the searchers and the fugitive, in other words, by placing restrictions on the arena
on which the game is played. Before we consider some examples, we introduce some
definitions and results relating to strategies.
5.1.1 Strategies
Since a graph searching game is a simple game, it follows that the winner is determined
by reachability, and therefore if either the fugitive or the searchers have a winning
strategy, they have a memoryless strategy. However, in this chapter we are interested in
resource bounded winning strategies, and in this case memoryless strategies, indeed,
even finite memory strategies may no longer be sufficient. However, to ensure that
computing such strategies remains decidable, we impose restrictions on the resource
measures we consider so that searches consistent with strategies are only ever simple
paths in the arena. This motivates the definition of a history-dependent strategy.
Definition 5.3 (History-dependent strategy). Let G be a graph, and GΓG a graph searching game on G defined by (Ls , Lf , A). Given a set Σ, a history-dependent strategy for
the searchers is a partial function σ : Σ∗ × Ls × Lf → Σ × Ls such that:
• σ(, X0 , R0 ) is defined for the empty word , and (X0 , R0 ) = vI (A),
• If σ(w, X, R) = (a, X 0 ) for (X, R) ∈ V0 (A), then
– (X, X 0 , R) ∈ V1 (A),
– there is an edge in E(A) from (X, R) to (X, X 0 , R), and
– if there is an edge in E(A) from (X, X 0 , R) to (X 0 , R0 ) ∈ V0 (A) then
σ(w · a, X 0 , R0 ) is defined.
We say a search π = (X0 , R0 )(X1 , R1 ) · · · is consistent with σ if there exists a word
w = a1 a2 · · · ∈ Σ∗ ∪ Σω such that for all i ≥ 0, Xi+1 = σ(a1 · · · ai , Xi , Ri ). We call
w the history consistent with π.
Remark. In the sequel we will usually define history-dependent strategies inductively,
often omitting the associated history when it is clear from the context what the play to
a given position should be.
Nevertheless, we show in Section 5.3 that the resource bounded strategies we are
primarily concerned with are equivalent to winning strategies in a graph searching
game. For this reason, we reserve the definition of strategies for positional strategies.
Definition 5.4 (Strategy). Let G be a graph, and GΓG a graph searching game on G
defined by (Ls , Lf , A). A strategy for the searchers is a partial function, σ : Ls ×
Lf → Ls , such that if σ(X, R) is defined there is an edge in E(A) from (X, R)
to (X, σ(X, R), R). If π = (X0 , R0 )(X1 , R1 ) · · · is a search in GΓG , we say π is
consistent with σ if for all i ≥ 0, Xi+1 = σ(Xi , Ri ). We say σ is winning (for the
searchers) if every search from vI (A) consistent with σ is winning for the searchers.
5.1. DEFINITIONS
73
A strategy for the fugitive is a partial function ρ : Ls × Ls × Lf such that if
(X, X 0 , R) ∈ V1 (A), there is an edge in E(A) from (X, X 0 , R) to (X 0 , ρ(X, X 0 , R)).
If π = (X0 , R0 )(X1 , R1 ) · · · is a search in GΓG , we say π is consistent with ρ if for all
i ≥ 0, Ri+1 = σ(Xi , Xi+1 , Ri ). We say ρ is winning (for the fugitive) if every search
from vI (A) consistent with ρ is winning for the fugitive.
Given a strategy σ for the searchers and a strategy ρ for the fugitive, the unique
maximal search consistent with σ and ρ is the search defined by σ and ρ
We now use strategies to define a structure that will prove useful in the next few
chapters. Given a strategy σ for the searchers in a graph searching game GΓG defined
by (Ls , Lf , A), we see that σ induces a subgraph of A in the following way. Let V ⊆
V (A) be the set of positions and intermediate positions reached by some play from
vI (A) consistent with σ. Considering for the moment positional strategies, it follows
that from each position (X, R) ∈ V there is precisely one successor (X, X 0 , R) ∈ V ,
namely the element of V1 (A) with X 0 = σ(X, R). The structure we are interested
in is a slight variation of this subgraph where, just as with our policy for describing
searches, the intermediate positions are ignored.
Definition 5.5 (Strategy digraph). Let G be a graph and GΓG a graph searching game on
G defined by (Ls , Lf , A). Let σ be a strategy for the searchers. The strategy digraph
defined by σ, Dσ , is the directed graph defined as:
• V (Dσ ) is the set of all pairs (X, R), including “positions” of the form (X, ∅),
such that there is some search in GΓG , (X0 , R0 )(X1 , R1 ) · · · , from vI (A) =
(X0 , R0 ) and consistent with σ, with (X, R) = (Xi , Ri ) for some i.
• There is an edge from (X, R) to (X 0 , R0 ) in E(Dσ ) if X 0 = σ(X, R) and either
there is an edge from (X, X 0 , R) to (X 0 , R0 ) in E(A), or there are no edges from
(X, X 0 , R) in E(A) and R0 = ∅.
Remark. Sometimes it may be convenient to assume that nodes of the form (X 0 , ∅) of
a strategy digraph are duplicated so that each such position actually corresponds to a
vertex (X, X 0 , R) in V1 (A). When this is the case, we see that every leaf of the form
(X, ∅) has a unique predecessor: if (X 0 , ∅) is associated with (X, X 0 , R) then (X, R)
is the unique predecessor of (X 0 , ∅). We observe that after these duplications, we still
have |V (Dσ )| ≤ |V (A)|.
An observation that will prove useful concerns the form the strategy digraph takes
for winning strategies.
Lemma 5.6. Let G be a graph and GΓG a graph searching game on G defined by
(Ls , Lf , A). If σ is a winning strategy for the searchers then Dσ is a directed acyclic
graph and all leaves of Dσ are of the form (X, ∅).
Proof. We observe that from the definition, there is a path from vI (A) = (X0 , R0 ) to
every node (X, R) ∈ V (Dσ ). We also observe that every path (X0 , R0 )(X1 , R1 ) · · ·
in Dσ from vI (A) corresponds to a search in GΓG consistent with σ, and if (X, R) is
a leaf then there is no search consistent with σ extending any consistent search which
ends at (X, R). Since σ is a winning strategy for the searchers, all searches consistent
74
CHAPTER 5. GRAPH SEARCHING GAMES
with σ can be extended to a complete search. Thus, if (X, R) is a leaf, it follows that
all searches from (X0 , R0 ) which end at (X, R) must be complete, so R = ∅. To show
acyclicity, it suffices to show that if Dσ is not acyclic, then σ is not a winning strategy
for the searchers. Suppose (Y1 , S1 ) · · · (Ym , Sm ) is a cycle in Dσ . By our earlier observation, π = (Y1 , S1 ) · · · (Ym , Sm )(Y1 , S1 ) is a search from (Y1 , S1 ) consistent with σ.
Now from the definition of V (Dσ ), there exists a search π 0 = (X0 , R0 ) · · · (Xk , Rk ),
where (Xk , Rk ) = (Y1 , S1 ) consistent with σ from (X0 , R0 ) = vI (A). Therefore, the
infinite search
π 0 · π · π · · · = (X0 , R0 ) · · · (Y1 , S1 ) · · · (Ym , Sm ), (Y1 , S1 ) · · ·
is a search from vI (A) consistent with σ. As this cannot possibly be extended to a
finite search and the fugitive wins all infinite plays, it follows that σ is not a winning
strategy for the searchers.
u
t
Definition 5.7 (Strategy DAG). Let G be a graph and GΓG a graph searching game on
G. If σ is a winning strategy for the searchers then we call Dσ the strategy DAG defined
by σ.
One important property of plays, searches and strategies that we are interested in
is the concept of monotonicity. In particular, we concentrate on two types of monotonicity: fugitive-monotonicity, where the set of potential fugitive locations is always
non-increasing, and searcher-monotonicity, where no location vacated by a searcher is
ever re-occupied.
Definition 5.8 (Fugitive and Searcher Monotonicity). Let G be a graph and let π =
(X0 , R0 )(X1 , R1 ) · · · be a search in a graph searching game on G. We say π is
• fugitive-monotone if Ri ⊇ Ri+1 for all i ≥ 0, and
• searcher-monotone if Xi ∩ Xk ⊆ Xj for 0 ≤ i ≤ j ≤ k.
A strategy, σ, for the searchers in a graph searching game on G is fugitive-monotone
(searcher-monotone) if every search consistent with σ is fugitive-monotone (searchermonotone).
Our next result concerning strategies in the general graph searching game is a useful
observation regarding monotone strategies. We show that, under some simple assumptions, a searcher-monotone winning strategy must also be fugitive-monotone. Let us
say that a graph searching game permits idling if the fugitive is able to remain at any
location which is not about to be occupied by a searcher. Furthermore, let us say that a
graph searching game is vacating sensitive if, whenever some location becomes available to the fugitive, there must be some location, previously occupied by a searcher,
that the fugitive can now occupy. More precisely,
Definition 5.9 (Permits idling). Let G be a graph and GΓG a graph searching game on
G defined by (Ls , Lf , A). We say GΓG permits idling if for all (X, X 0 , R) ∈ V1 (A)
and all r ∈ R \ X 0 , there exists R0 ⊆ Elts(G) such that r ∈ R0 and there is an edge in
E(A) from (X, X 0 , R) to (X 0 , R0 ).
5.1. DEFINITIONS
75
Definition 5.10 (Vacating sensitive). Let G be a graph and GΓG a graph searching game
on G defined by (Ls , Lf , A). We say that GΓG is vacating sensitive if, whenever there
is an edge in E(A) from (X, X 0 , R) to (X 0 , R0 ) with R0 6⊆ R, then X ∩ R0 6= ∅.
Lemma 5.11. Let G be a graph and GΓG a graph searching game on G which permits
idling and is vacating sensitive. If σ is a searcher-monotone winning strategy for the
searchers on GΓG , then σ is fugitive-monotone.
Proof. Suppose π = (X0 , R0 )(X1 , R1 ) · · · is a search consistent with σ which is not
fugitive-monotone. Let i be the least index such that Ri 6⊇ Ri+1 . Since GΓG is vacating
sensitive, there exists r ∈ Xi ∩ Ri+1 . But then, as GΓG permits idling, the fugitive can
always choose a response which includes r until it is occupied by a searcher. That is,
there is a search π 0 = (X00 , R00 )(X10 , R10 ) · · · , consistent with σ, which agrees with π up
to (Xi+1 , Ri+1 ) and either there is some k such that r ∈ Rj0 for all j with i+1 ≤ j < k
and r ∈ Xk , or r ∈ Rj0 for all j ≥ i + 1. In the first case, we have r ∈ Xi0 ∩ Xk0 but as
0
0
r ∈ Ri+1
, we also have r ∈
/ Xi+1
, contradicting the fact that σ is searcher-monotone.
0
In the second case, since Rj 6= ∅ for all j, it follows that π 0 is an infinite search,
contradicting the fact that σ is a winning strategy for the searchers.
u
t
Remark. Earlier, we asserted that variations of graph searching games are obtained
by imposing restrictions on the arena. In this way, we see that questions relating to
fugitive-monotone strategies can be viewed as questions in a restricted version of the
game: the game defined in the same way with the restriction that we do not allow
the searchers to make any move which enables the fugitive to make a non-monotone
move (a move for which the set of potential fugitive locations is not non-increasing).
That is, if A is the arena of a graph searching game, let A0 be the arena obtained
by removing edges from (X, R) to (X, X 0 , R) if there is an edge from (X, X 0 , R) to
(X 0 , R0 ) where R0 6⊆ R. Now a strategy for the searchers on A0 is a fugitive-monotone
strategy for the searchers on A. On the other hand, searcher-monotonicity is a more
dynamic restriction – the moves available to the searchers are dependent on the play to
that point. Lemma 5.11 illustrates how, in some cases, the strategy restrictions imposed
by searcher-monotonicity can also be interpreted as restrictions on the game.
5.1.2 Simulations
In Definition 2.20, we saw the idea of a game simulation. We now introduce a refinement of this suitable for graph searching games.
Definition 5.12 (Searching simulation). Let GΓG be a graph searching game on G de0
fined by (Ls , Lf , A), and GΓG 0 be a graph searching game on G 0 defined by (L0s , L0f , A0 ).
0
A searching simulation from GΓG to GΓG 0 is a pair of relations (Rs , Rf ) such that:
• Rs ⊆ Ls × L0s , Rf ⊆ Lf × L0f , and
• The relation S on V (A) × V (A0 ) defined by
– (X, R) S (Y, R0 ) if (X, Y ) ∈ Rs and (R, R0 ) ∈ Rf , and
– (X, X 0 , R) S (Y, Y 0 , R0 ) if (X, Y ), (X 0 , Y 0 ) ∈ Rs and (R, R0 ) ∈ Rf ,
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CHAPTER 5. GRAPH SEARCHING GAMES
is a game simulation from A to A0 .
As a searching simulation is a restricted game simulation, and searches correspond
to plays in the arena, the next result follows immediately from Lemma 2.21.
0
Lemma 5.13. Let GΓG be a graph searching game on G defined by (Ls , Lf , A), and GΓG 0
be a graph searching game on G 0 defined by (L0s , L0f , A0 ). Let (Rs , Rf ) be a searching
0
simulation from GΓG to GΓG 0 with (∅, ∅) ∈ Rf . For all searcher strategies σ on GΓG
0
0
and all fugitive strategies ρ0 on GΓG 0 , there exists a searcher strategy σ 0 on GΓG 0 and
Γ
a fugitive strategy ρ on GG such that if π(σ,ρ) = (X1 , R1 )(X2 , R2 ) · · · is the search
in GΓG defined by σ and ρ, and π(σ0 ,ρ0 ) = (X10 , R10 )(X20 , R20 ) · · · is the search in GΓG
defined by σ 0 and ρ0 , then |π(σ,ρ) | = |π(σ0 ,ρ0 ) |, and (Xi , Xi0 ) ∈ Rs and (Ri , Ri0 ) ∈ Rf
for all i ≤ |π(σ,ρ) |.
As with game simulations, we observe that the definition of the strategy σ 0 is independent of the choice of ρ. This gives us the following analogue to Corollary 2.22:
0
Corollary 5.14. Let GΓG be a graph searching game on G, and GΓG 0 be a graph search0
ing game on G 0 . Let (Rs , Rf ) be a searching simulation from GΓG to GΓG 0 with (∅, ∅) ∈
Rf , and let σ be a strategy for the searchers on GΓG . Then there exists a strategy σ 0 for
0
the searchers on GΓG 0 such that for every search (X10 , R10 )(X20 , R20 ) · · · consistent with
σ 0 there exists a search (X1 , R1 )(X2 , R2 ) · · · consistent with σ with (Xi , Xi0 ) ∈ Rs
and (Ri , Ri0 ) ∈ Rf for all i ≥ 1.
As with game simulations, we call the strategies which we can derive from a simulation simulated strategies.
Definition 5.15 (Simulated search strategy). The strategy σ 0 in Corollary 5.14 is called
a strategy (Rs , Rf )-simulated by σ.
This enables us to state the following consequence of Corollary 2.27.
0
Lemma 5.16. Let GΓG be a graph searching game on G and GΓG 0 a graph searching
0
game on G 0 . Let (Rs , Rf ) be a searching simulation from GΓG to GΓG 0 , and let σ be a
0
strategy for the searchers on GΓG . If σ 0 is a strategy (Rs , Rf )-simulated by σ on GΓG 0 ,
then:
1. If σ is a winning strategy, then σ 0 is a winning strategy, and
2. If (X, X 0 ) ∈ Rs and (R, R0 ) ∈ Rf , then σ(X, R), σ 0 (X 0 , R0 ) ∈ Rs .
With some straightforward assumptions about the relations which comprise a searching simulation, we can show that strategies simulated by monotone strategies are also
monotone. First we recall two definitions regarding relations of sets.
Definition 5.17 (Monotone and ∩-compatible relation). Let X and Y be sets, and let
R ⊆ P(X) × P(Y ) be a relation between subsets of X and subsets of Y . We say R is
monotone if for all (A, A0 ), (B, B 0 ) ∈ R with A ⊆ B, we have A0 ⊆ B 0 . We say R is
∩-compatible if for all (A, A0 ), (B, B 0 ) ∈ R, (A ∩ B, A0 ∩ B 0 ) ∈ R.
5.1. DEFINITIONS
77
0
Lemma 5.18. Let GΓG be a graph searching game on G and GΓG 0 a graph searching
0
game on G 0 . Let (Rs , Rf ) be a searching simulation from GΓG to GΓG 0 , and let σ is a
0
strategy for the searchers on GΓG . If σ 0 is a strategy (Rs , Rf )-simulated by σ on GΓG 0 ,
then:
1. If Rf is monotone and σ is fugitive-monotone, then σ 0 is fugitive-monotone, and
2. If Rs is monotone and ∩-compatible and σ is searcher-monotone, then σ 0 is
searcher-monotone.
Proof. Let π 0 = (X10 , R10 )(X20 , R20 ) · · · be a search consistent with σ 0 . By the definition of simulated strategies, there exists a search (X1 , R1 ) · · · consistent with σ such
that (Xi , Xi0 ) ∈ Rs and (Ri , Ri0 ) ∈ Rf for all i ≥ 1.
1: If σ is fugitive-monotone, then Ri ⊇ Ri+1 for all i ≥ 1, so if Rf is monotone,
0
it follows that Ri0 ⊇ Ri+1
for all i ≥ 1. Thus π 0 is fugitive monotone, and as π 0 was
0
arbitrary, it follows that σ is fugitive-monotone.
2: If σ is searcher-monotone, then for all i ≤ j ≤ k, we have Xi ∩ Xk ⊆ Xj . If
Rs is ∩-compatible, then (Xi ∩ Xk , Xi0 ∩ Xk0 ) ∈ Rs , and so if Rs is also monotone,
then Xi0 ∩ Xk0 ⊆ Xj0 . Thus π 0 is searcher-monotone, and as π 0 was arbitrary, it follows
that σ 0 is searcher-monotone.
u
t
We now introduce some concepts that will prove useful later when we establish
robustness results for graph searching games.
Definition 5.19 (Quasi-simulation family). A quasi-simulation family is a partial function R which assigns to a pair of graphs (G, G 0 ) a pair of relations (Rs0 , Rf0 ) with
Rs0 , Rf0 ⊆ P(Elts(G)) × P(Elts(G 0 )).
Often it is easier to define a quasi-simulation family as a pair of partial functions
(Rs , Rf ), each of which takes a pair of graphs (G, G 0 ) to a relation from P(Elts(G))
to P(Elts(G 0 ))
Definition 5.20 (R-closure). Let R be a quasi-simulation family, and Γ a graph searching type. We say Γ is R-closed if for any pair of graphs G and G 0 with R(G, G 0 ) =
(Rs0 , Rf0 ), Γ(G) = (Ls , Lf , A) and Γ(G 0 ) = (L0s , L0f , A0 ); (Rs , Rf ) is a searching
simulation from GΓG 0 to GΓG , where Rs = Rs0 ∩ (L0s × Ls ) and Rf = Rf0 ∩ (L0f × Lf ).
To help gain an intuition, we provide an example of R-closure. Consider the following property of graph searching game types.
Definition 5.21 (Respects restriction). Let Γ be a graph searching game type. We say
Γ respects restriction if for any graphs G and G 0 such that G is a subgraph of G 0 , if
Γ(G) = (Ls , Lf , A) and Γ(G 0 ) = (L0s , L0f , A0 ), then
0
• If Rmax is the ⊆-maximal element of Lf , and Rmax
is the ⊆-maximal element
0
0
of Lf , then Rmax = Rmax ∩ Elts(G).
• If there is an edge from (X, R) to (X, X 0 , R) in E(A0 ) and v = (X∩Elts(G), R∩
Elts(G)) ∈ V (A), then there is an edge from v to (X ∩Elts(G), X 0 ∩Elts(G), R∩
Elts(G)) in E(A), and
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CHAPTER 5. GRAPH SEARCHING GAMES
• If there is an edge from (Y, Y 0 , S) to (Y 0 , S 0 ) in E(A) then for all X, X 0 , R
such that (X, X 0 , R) ∈ V1 (A0 ), Y = X ∩ Elts(G), Y 0 = X 0 ∩ Elts(G), and
S = R ∩ Elts(G), there exists R0 such that S 0 = R0 ∩ Elts(G) and there is an
edge from (X, X 0 , R) to (X 0 , R0 ) in E(A0 ).
Intuitively, if a graph searching game type respects restriction, then if G is a subgraph of G 0 , a strategy for the searchers in G 0 is also a strategy in G when we disregard
the elements of G 0 which are not part of G. In other words, a restriction of a search
strategy is a search strategy of a restriction. In Section 5.4 we introduce the dual notion, restriction reflection, in which a search strategy of a graph can be viewed as a
search strategy in any larger graph. We now show that this property corresponds to an
R-closure for a quasi-simulation family R of relations similar to the superset relation.
Definition 5.22 (⊃
· ). For each pair of graphs (G 0 , G), with G a subgraph of G 0 , we
G0
define ⊃
· G ⊆ P(Elts(G 0 )) × P(Elts(G)) as follows. For A ⊆ Elts(G 0 ) and B ⊆ Elts(G)
0
we say A ⊃
· GG B if B = A ∩ Elts(G). Let ⊃
· denote the function which assigns to each
0
0
0
pair of graphs (G , G), with G a subgraph of G 0 , the pair of relations (⊃
· GG , ⊃
· GG ).
Lemma 5.23. Let Γ be a graph searching game type. Then Γ respects restriction if,
and only if, Γ is ⊃
· -closed.
Proof. Let G and G 0 be graphs. We observe that if neither G is a subgraph of G 0 nor
G 0 is a subgraph of G then nothing can be said about whether Γ respects restriction or
whether Γ is ⊃
· -closed. Thus we assume without loss of generality that G is a subgraph
of G 0 . Let Γ(G) = (Ls , Lf , A) and Γ(G 0 ) = (L0s , L0f , A0 ). For convenience we will
0
drop the subscript and superscript and use ⊃
· to denote the relation ⊃
· GG .
First let us assume Γ respects restriction. From the definition of ⊃
· , we have
Elts(G 0 ) ⊃
· Elts(G), thus we must show that (⊃
· ,⊃
· ) is a searching simulation from
GΓG 0 to GΓG . In the definition of R-closure, we assume ⊃
· is restricted to be a relation on
the appropriate sets, so it suffices to show that the relation defined by pointwise application of ⊃
· is a game simulation from A0 to A. For convenience we will also denote
the pointwise relation by ⊃
· . Clearly, since ∅ ∩ Elts(G) = ∅, we have ∅ ⊃
· ∅. Further0
is the ⊆-maximal element
more, if Rmax is the ⊆-maximal element of Lf and Rmax
0
0
of L0f , then as Γ respects restriction, Rmax = Rmax
∩ Elts(G). Thus Rmax
⊃
· Rmax ,
0
and (∅, Rmax ) ⊃
· (∅, Rmax ). Thus (⊃
· ,⊃
· ) satisfies (SIM-1). Now suppose there is
an edge from (X, R) to (X, X 0 , R) in A0 and (X, R) ⊃
· (Y, S). From the definition
of ⊃
· , Y = X ∩ Elts(G) and S = R ∩ Elts(G), so by the definition of respecting
restriction, there is an edge from (Y, S) to (Y, X 0 ∩ Elts(G), S) in A. Since clearly
X0 ⊃
· (X 0 ∩ Elts(G)), it follows that (SIM-2) is satisfied. Finally suppose there is an
edge in A from (Y, Y 0 , S) to (Y 0 , S 0 ) and (Y, Y 0 , S) ⊃
· (X, X 0 , R). From the defi0
0
nition of ⊃
· , we have Y = X ∩ Elts(G), Y = X ∩ Elts(G) and S = R ∩ Elts(G).
Thus, as Γ respects restriction, there exists R0 ∈ L0f such that S 0 = R0 ∩ Elts(G) and
there is an edge in A from (X, X 0 , R) to (X 0 , R0 ). Since X 0 ⊃
· Y 0 and R0 ⊃
· S 0,
0
0
0
0
it follows that (X , R ) ⊃
· (Y , S ), thus (SIM-3) is satisfied. Therefore, (⊃
· ,⊃
· ) is a
searching simulation from GΓG 0 to GΓG . Since G and G 0 were arbitrary, it follows that Γ
is ⊃
· -closed.
5.2. EXAMPLES
79
Now suppose Γ is ⊃
· -closed. Since the relation defined by pointwise application of
0
⊃
· is a game simulation from A0 to A, vI (A) = (∅, Rmax ), and vI (A0 ) = (∅, Rmax
),
0
it follows from (SIM-1) that ∅ ⊃
· ∅ and Rmax
⊃
· Rmax . From the definition of ⊃
· , it
0
follows that Rmax = Rmax
∩ Elts(G). Now suppose there is an edge from (X, R) to
(X, X 0 , R) in A0 , and (Y, S) ∈ V (A) where Y = X ∩ Elts(G) and S = R ∩ Elts(G).
From the definition of ⊃
· , it follows that X ⊃
· Y and R ⊃
· S, thus as (⊃
· ,⊃
· ) is a
game simulation, it follows from (SIM-2) that there exists v 0 such that there is an
edge from (Y, S) to v 0 and v 0 is related to (X, X 0 , R) by the pointwise application of
⊃
· . By the definition of graph searching games, v 0 = (Y, Y 0 , S) for some Y 0 ∈ Ls ,
and by the definition of searching simulation X 0 ⊃
· Y 0 . Thus Y 0 = X 0 ∩ Elts(G).
0
Finally suppose there is an edge from (Y, Y , S) to (Y 0 , S 0 ) and X, X 0 , R are such that
Y = X ∩ Elts(G), Y 0 = X 0 ∩ Elts(G) and S = R ∩ Elts(G). From the definition of ⊃
·,
X⊃
· Y , X0 ⊃
· Y 0 and R ⊃
· S. Thus, from (SIM-3), there exists v ∈ V0 (A0 ) such that
there is an edge from (X, X 0 , R) to v and v is related to (Y 0 , S 0 ). From the definition of
graph searching games, v = (X 0 , R0 ) for some R0 , and by the definition of searching
simulation, R0 ⊃
· S 0 . Thus S 0 = R0 ∩ Elts(G). Therefore, all conditions necessary
for respecting restriction are satisfied. Since G and G 0 were arbitrary, it follows that Γ
respects restriction.
u
t
5.2 Examples
We now look at some examples of graph searching game types which occur in the
literature. Many of these examples were introduced to provide an intuitive understanding of some of the graph parameters we discussed in the previous chapter. We show
how each of these games can be described using the framework we have introduced,
thereby motivating the use of graph searching games to formally define measures of
graph complexity.
5.2.1 Cops and visible robber
The cops and visible robber game was introduced in [ST93] to provide a characterization of tree-width. We can define it as a graph searching game played on an undirected
graph G, as follows.
Definition 5.24 (Cops and visible robber game). Let G be an undirected graph. The
cops and visible robber game on G is a graph searching game on G defined by the triple
(Ls , Lf , A) where:
• Ls = P(V (G)), Lf = {R ⊆ V (G) : R is non-empty and connected}∪{V (G)},
• (X, R) ∈ V0 (A) if R is a connected component of G \ X,
• (X, X 0 , R) ∈ V1 (A) if (X, R) ∈ V0 (A) and X 0 ∈ Ls ,
• (X, R), (X, X 0 , R) ∈ E(A) for all (X, R) ∈ V0 (A),
• (X, X 0 , R), (X 0 , R0 ) ∈ E(A) if R∪R0 is contained in a connected component
of G \ (X ∩ X 0 ).
80
CHAPTER 5. GRAPH SEARCHING GAMES
Intuitively, the cops (searchers) and robber (fugitive) occupy vertices of the graph.
There is no constraint on the cops, they can be removed and placed on any set of
vertices. The robber is constrained to move along paths of any length in the graph,
provided he does not pass through a stationary cop. The robber’s location in the graph
is known to the cops, but because he is able to move infinitely fast, we view his set of
potential locations as a connected component of the subgraph obtained by removing
vertices occupied by cops. A move consists of some cops being removed from the
graph, and announcing vertices that are about to be occupied. The robber is then able
to move to any vertex he can reach, and then cops are placed on the announced vertices.
If the robber is located on a vertex which has become occupied, then he is captured and
the cops win. If he can avoid capture forever, then he wins.
We observe that the cops and visible robber game permits idling: given an intermediate position (X, X 0 , R) and r ∈ R \ X 0 , let R0 be the connected component of G \ X 0
which contains r. Then R ∪ R0 is contained in a connected component as they are
connected sets with a non-empty intersection. Thus there is an edge from (X, X 0 , R)
to (X 0 , R0 ). Furthermore, the game is vacating sensitive: if it is possible to move from
(X, X 0 , R) to (X 0 , R0 ) where R0 6⊆ R then there exists r ∈ R0 \ R such that r is
adjacent to some vertex in R. Now R ∪ {r} is connected, so if r ∈
/ X, then R is not
a connected component of G \ X. Hence r ∈ X, so X ∩ R0 6= ∅. Thus we can apply
Lemma 5.11 to obtain:
Lemma 5.25. A cop-monotone winning strategy in the cops and visible robber game
is robber-monotone.
There are some interesting variants of the cops and visible robber game obtained by
restricting the movements of the cops. For example, cops are either removed or placed
so (X, X 0 , R) is an intermediate position only if either X 0 ⊆ X, or X ⊆ X 0 ; at most
one cop is moved, so (X, X 0 , R) is an intermediate position only if |X 0 4 X| ≤ 1; or
at most one cop is placed, so (X, X 0 , R) is an intermediate position only if |X 0 \ X| ≤
1. Another variation is the following parameterized class of games, in which we bound
the number of cops trying to capture the robber:
Definition 5.26 (k-cops and visible robber game). Let G be an undirected graph. The
k-cops and visible robber game on G is defined as the cops and visible robber game,
except Ls = [V (G)]≤k .
In Section 5.3 we show that strategies in these games are equivalent to resourcebounded strategies in the unrestricted game, where the resource we are concerned with
is the maximum number of cops occupying the graph at any stage. While this may
seem obvious, the observation is quite useful when we consider the complexity of the
problem of determining the existence of resource-bounded winning strategies.
We also show in Section 5.3 how this game, particularly this last variant, is closely
connected to tree-width. So it would seem that extending this game to directed graphs
would be a useful way to generalize tree-width to directed graphs. There are two obvious ways to extend this game: we could extend the informal description, constraining
the robber to move along directed paths of any length; or we could extend the formal
description, having positions (X, R) where R is a strongly connected component of
5.2. EXAMPLES
81
G \ X, and a transition from (X, R) to (X 0 , R0 ) if R ∪ R0 is contained in a strongly
connected component of G \ (X ∩ X 0 ). The game corresponding to the latter extension seems less intuitive: it corresponds to restricting the robber to being able to move
along directed paths to any vertex from which he has a directed cop-free path back
to his starting vertex. This game, which we call the strongly connected visible robber
game, or more simply the strong visible robber game, was considered in [JRST01], and
later in this chapter we discuss its relationship with directed tree-width. We investigate
the other, arguably more natural, generalization in Chapter 6.
5.2.2 Cops and invisible robber
The cops and invisible robber game, also known as the node searching game, or vertex
decontamination has been well-studied in the context of graph theory [KP86, BS91,
LaP93]. In our framework, the definition is as follows.
Definition 5.27 (Cops and invisible robber game). Let G be an undirected graph. The
cops and invisible robber game on G is a graph searching game on G defined by the
triple (Ls , Lf , A) where:
• Ls = P(V (G)), Lf = P(V (G)) \ {∅},
• (X, R) ∈ V0 (A) if R is a union of non-empty connected components of G \ X,
• (X, X 0 , R) ∈ V1 (A) if (X, R) ∈ V0 (A) and X 0 ∈ Ls ,
• (X, R), (X, X 0 , R) ∈ E(A) for all (X, R) ∈ V0 (A),
• (X, X 0 , R), (X 0 , R0 ) ∈ E(A) if R0 = ReachG\(X∩X 0 ) (R) \ X 0 .
The game is played on an undirected graph G in the same way as the cops and
visible robber: the cops are free to move anywhere on G, and the robber can run at
great speed along cop-free paths in the graph. In this game however, the location of
the robber is not known to the cops – they are only aware of the vertices the robber
cannot be at: either because those vertices are currently occupied by cops, or there is
no possibility that the robber could not have reached those vertices from when they
were vacated by cops. So positions in this game are pairs (X, R) where X, R ⊆
V (G) and R is a union of connected components of G \ X, and a search in this game
ending at (X, R) can be extended to a search ending at (X 0 , R0 ) if, and only if, R0 =
ReachG\(X∩X 0 ) (R). We observe that since R0 is uniquely determined from X, X 0 and
R, the robber has no choice from the intermediate position (X, X 0 , R), so this game is
effectively a single player game.
In the literature, this game is often viewed as the problem of trying to clean a
contaminated graph. Vertices where the robber could be are “contaminated”, vertices
where the robber cannot be are “cleared”, and occupation of a vertex by a cop “clears”
that vertex.
CHAPTER 5. GRAPH SEARCHING GAMES
82
5.2.3 Cave searching
The next game we consider is an example of a searching game motivated by a real-life
problem. In [Bre67], in a publication for the spelunking community, Breisch considered the problem of finding a lost person in a cave system. In response to a question
posed by some cavers about whether existing search techniques could be improved,
Parsons [Par78] reformulated the problem as a graph-theoretical problem and investigated games known as graph sweeping games. These can be defined as graph searching
games as follows.
Definition 5.28 (Graph sweeping game). Let G be an undirected graph. The graph
sweeping game on G is the graph searching game on G defined by the triple (Ls , Lf , A),
where:
• X ∈ Ls if, and only if, X = V ∪ E, where V ⊆ V (G), E ⊆ E(G), |E| ≤ 1,
and if E = {e} then e ∩ V 6= ∅,
• Lf = P(Elts(G)) \ {∅},
• (X, R) ∈ V0 (A) if, and only if, X ∩ R = ∅,
• (X, X 0 , R) ∈ V1 (A) if, and only if, X = V ∪ E, X 0 = V 0 ∪ E 0 , withV, V 0 ⊆
V (G) and E, E 0 ⊆ E(G), and either E 0 = ∅ and V 0 \V = ∅, or if E 0 = {u, v}
with v ∈ V 0 then u ∈ V .
• If (X, R) ∈ V0 (A) and (X, X 0 , R) ∈ V1 (A) then (X, R), (X, X 0 , R)) ∈
E(A), and
• There is an edge from (X, X 0 , R) to (X 0 , R0 ) if, and only if, R0 consists of
0
all elements x ∈ Elts(G)
\ X such that if C is the connected component of
G \ X 0 ∩ (X ∪ E(G)) which contains x, then C ∩ R 6= ∅.
In this game, the graph represents the cave system, with edges representing traversable
paths. The fugitive, or lost caver, is located somewhere in the cave system – represented in this game by having sets of elements of G for the locations of the fugitive.
The searchers move through the graph by moving from one vertex to an adjacent vertex
along an edge connecting them.
5.2.4 Detectives and robber
The next game was introduced by Berwanger and Grädel [BG04] to define a measure of
complexity for directed graphs known as entanglement. We can present their definition
in terms of graph searching games as follows.
Definition 5.29 (Detectives and robber game). Let G be a directed graph. The detectives and robber game on G is a graph searching game defined by the triple (Ls , Lf , A)
where:
• Ls = P(V (G)), Lf = {r} : r ∈ V (G)} ∪ {V (G)},
5.2. EXAMPLES
83
• V0 (A) = {(∅, V (G))} ∪ {(X, {r}) : r ∈
/ X},
• V1 (A) = {(∅, ∅, V (G))} ∪ {(X, X 0 , {r}) : (X, {r}) ∈ V0 (A) and X 0 ⊆ X ∪
{r}},
• If (X, R) ∈ V0 (A) and (X, X 0 , R) ∈ V1 (A) then (X, R), (X, X 0 , R) ∈
E(A),
• There is an edge from (∅, ∅, V (G)) to (∅, {r}) for all r ∈ V (G),
• For all (r, r0 ) ∈ E(G) and (X, X 0 , {r}) ∈ V1 (A) with r0 ∈
/ X 0 , there is an edge
0
0 0
in E(A) from (X, X , {r}) to (X , r ), and
• There are no other edges in E(A).
In this game, the detectives and robber occupy vertices in the graph. The robber
has to move to a successor of his current location and the detectives can only move to
the last position of the robber or remain where they are.
5.2.5 Cops and inert robber
As with the cops and visible robber game defined in Definition 5.24, the final game
we consider is also a game played on an undirected graph closely related to tree-width.
Introduced by Dendris, Kirousis and Thilikos [DKT97], the cops and inert robber game
can also be viewed as a graph searching game in the following manner.
Definition 5.30 (Cops and inert robber). Let G be an undirected graph. The cops
and inert robber game on G is the graph searching game on G defined by the triple
(Ls , Lf , A), where:
• Ls = P(V (G)), Lf = P(V (G)) \ {∅},
• (X, R) ∈ V0 (A) if R is a union of non-empty connected components of G \ X,
• (X, X 0 , R) ∈ V1 (A) if (X, R) ∈ V0 (A) and X 0 ∈ Ls ,
• (X, R), (X, X 0 , R) ∈ E(A) for all (X, R) ∈ V0 (A),
• (X, X 0 , R), (X 0 , R0 ) ∈ E(A) if R0 = R ∪ ReachG\(X∩X 0 ) (R ∩ X 0 ) \ X 0 .
As with the cops and invisible robber game defined in Definition 5.27, in this game
the cops and robber occupy vertices of the graph, the cops are free to move anywhere in
the graph, and the robber may run at great speed along paths in the graph. Furthermore,
the location of the robber is unknown to the cops. However we impose the restriction
that he is only able to move from his position if it is about to be occupied by a cop.
Thus at position (X, R), X represents the location of the cops and R represents the
set of potential locations. Now if the cops move to X 0 , then the resulting potential
locations for the robber consist of his current set of locations together with any vertex
84
CHAPTER 5. GRAPH SEARCHING GAMES
v for which there is a path from a vertex in R ∩ X 0 to v, excluding any vertex now
occupied by a cop. Thus R0 , the new set of potential locations, can be defined as:
R0 = R ∪ ReachG\(X∩X 0 ) (R ∩ X 0 ) \ X 0 .
In the next section we see that this game is also closely connected to tree-width,
suggesting that the generalization of this game to directed graphs would be a practical way to develop complexity measures which extend tree-width. In Chapter 7 we
consider such a generalization.
5.2.6 Cops and robber games
Examples 5.2.1, 5.2.2, and 5.2.5 highlight one of the most important and simple variants of the graph searching game, the cops and robber game. In this game the cops
(searchers) and the robber (fugitive) only occupy vertices of graph, with the robber
being able to start at any vertex of the graph.
Definition 5.31 (Cops and robber game). Let G be a graph and GΓG be a graph searching
game on G defined by a triple (Lc , Lr , A). We say GΓG is a cops and robber game if
Lc ⊆ P(V (G)), Lr ⊆ P(V (G)) and V (G) ∈ Lr . We call the searchers of a cops
and robber game the cops, and the fugitive is called the robber. Likewise, searchermonotone searches and strategies are cop-monotone and fugitive-monotone searches
and strategies are robber-monotone. A graph searching game type Γ is a cops and
robber game type if for all graphs G, GΓG is a cops and robber game.
One advantage of the restriction of the searchers and fugitive to vertices of the graph
is that the resulting games are less dependent on the edges of the graph. In particular, it
is often the case that the presence of multiple edges or loops does not affect the game
– the arena is the same as the arena for the graph searching game on the graph with
all loops removed and all multiple edges replaced with a single edge. In the sequel
we assume all cops and robber games are played on simple graphs, unless otherwise
stated.
5.3 Complexity measures
Unlike the games we considered in Chapter 2, we are not solely concerned with which
player wins a graph searching game. In most of the examples above, it is clear that the
searchers can always find the fugitive by (eventually) occupying all of the graph, so as
it stands the question is not interesting – the searchers always have a winning strategy.
One exception to this is the parameterized class of games, the k-cops and visible robber
games defined in Definition 5.24. This suggests that it may be more fruitful to consider
resource-bounded strategies. For instance, for a cops and robber game, we can ask
“Given k ∈ N, can the cops capture the robber while at any time occupying at most k
vertices?”. Consistent with viewing the cops as physical entities, this can be viewed as
asking if there is a winning strategy for k cops, defined more precisely as:
5.3. COMPLEXITY MEASURES
85
Definition 5.32 (Winning strategy for k cops). Let GΓG be a cops and robber game, σ
a strategy for the cops, and k ∈ N. We say that σ is a winning strategy for k cops if
σ is a winning strategy, and for any search (X0 , R0 )(X1 , R1 ) · · · consistent with σ,
|Xi | ≤ k for all i.
From this we can derive a complexity measure, in this particular case, the minimum
number of cops required to capture the robber. In the following chapters this is the
measure we are interested in, but for the remainder of this chapter we consider a more
general framework which encompasses many other important graph parameters. For
this we introduce the concept of a resource measure that can be used to restrict plays
and, by association, strategies in a graph searching game. First, we introduce two
partial orders on the class of sequences of sets.
Definition 5.33. Let π = X1 X2 · · · and π 0 = Y1 Y2 · · · be two (possibly infinite)
sequences of sets. We write π 0 ≤ π if π 0 is a subsequence of π. That is, there exists
an increasing sequence of indices n1 < n2 < · · · ≤ |π| such that Yi = Xni for all
i ≤ |π 0 |. We write π 0 ⊆ π if |π 0 | ≤ |π| and for all i ≤ |π 0 |, Yi ⊆ Xi .
Definition 5.34 (Resource measure). A resource measure is a function ϕ which maps
sequences of finite sets to elements of ω ∪ {ω}, with ϕ(π) = ω only if π is infinite.
We say ϕ is order-preserving (order-reversing) if for all π, π 0 ∈ dom(ϕ), π 0 ≤ π ⇒
ϕ(π 0 ) ≤ ϕ(π) (π 0 ≤ π ⇒ ϕ(π 0 ) ≥ ϕ(π)). We say ϕ is monotone (anti-monotone) if
for all π, π 0 ∈ dom(ϕ), π 0 ⊆ π ⇒ ϕ(π 0 ) ≤ ϕ(π) (π 0 ⊆ π ⇒ ϕ(π 0 ) ≥ ϕ(π)).
The resource measure which motivated the above discussion is an example of a
monotone, order-preserving resource measure:
Definition 5.35 (ϕmax ). The resource measure ϕmax is defined as follows. If π =
X1 X2 · · · is a sequence of finite sets, then
ϕmax (π) = max{|Xi |}.
i≥1
A resource measure ϕ defines a measure on a search π = (X0 , R0 )(X1 , R1 ) · · · in
the following way: let π1 = X0 X1 · · · be the sequence of first components of elements
of π, and define ϕ(π) := ϕ(π1 ). We only consider the sequence of searcher locations
because we are primarily interested in the resource usage of the searchers. It follows
that requiring a resource measure to be bounded imposes a restriction on the searches,
and consequently, the strategies available in a graph searching game. So asking if the
searchers have a winning strategy is no longer a trivial problem. Indeed, it would
seem that interesting metrics for graphs could be derived from the “optimal” bounds
of resource measures for which the searchers still have a winning strategy. This leads
to the following definition of a very general measure of graph complexity defined by
graph searching games.
Definition 5.36 (Graph searching width). Let Γ be a graph searching game type, and
ϕ an order-preserving (order-reversing) resource measure. Let G be a graph. The
(Γ, ϕ)-width of G, w(Γ,ϕ) (G), is the minimum (maximum) k such that in GΓG there
exists a winning strategy for the searchers, σ, so that for any search, π, consistent
CHAPTER 5. GRAPH SEARCHING GAMES
86
with σ, we have ϕ(π) ≤ k (ϕ(π) ≥ k). Likewise, if we restrict to fugitive-monotone
or searcher-monotone winning strategies in GΓG , we obtain the fugitive-monotone or
searcher-monotone (Γ, ϕ)-width of G.
Remark. As we are interested in minimizing (maximizing) an order-preserving (orderreversing) measure, it suffices to consider searches that are simple paths in the arena
– any loops are only going to increase (decrease) the resource requirements. Consequently, we only need to consider strategies that require finite memory to determine if
the searchers have a resource bounded winning strategy. Thus, the requirement that the
resource measure is order-preserving (or order-reversing) ensures that the restriction of
searches obtained by bounding the resource measure does not affect the decidability
of determining if the searchers have a winning strategy. In particular, the requirement
maintains our maxim that strategies with finite memory are sufficient, especially for
the resource bounded game.
Many practical measures of graph complexity can be defined using this framework,
as we see with the following examples.
5.3.1 Example: Cops and visible robber
We recall the cops and visible robber game defined in Example 5.2.1. In [ST93] when
this game was first considered, Seymour and Thomas showed that it could be used to
characterize tree-width by observing that the number of cops required to capture the
robber was equal to one more than the tree-width of the graph being searched. More
precisely, they proved:
Theorem 5.37 ([ST93]). Let G be an undirected graph. The following are equivalent:
1. G has tree-width ≤ k − 1.
2. k cops have a cop-monotone winning strategy in the cops and visible robber
game.
3. k cops have a robber-monotone winning strategy in the cops and visible robber
game.
4. k cops have a winning strategy in the cops and visible robber game.
Recalling the definition of ϕmax in Definition 5.35, we can rephrase this theorem as:
Corollary 5.38. Let Γ be the cops and visible robber game type defined in Definition 5.24, and let G be an undirected graph. Then
Tree-width(G) = w(Γ,ϕmax ) (G).
We remarked in Example 5.2.1 that there were several variants of the cops and
visible robber depending on various restrictions placed on the movement of the cops.
It is easy to see informally that the number of cops required to catch the robber in each
of these games is the same. We now provide a formal proof of this often glossed-over
point.
5.3. COMPLEXITY MEASURES
87
Proposition 5.39. Let Γ0 be the cops and visible robber game type defined in Definition 5.24. Let Γ1 be the cops and visible robber game type where cops are either placed
or removed. Let Γ2 be the cops and visible robber game type where at most one cop is
placed, and let Γ3 be the cops and visible robber game type where at most one cop is
moved at a time. Let G be an undirected graph. Then the following are equivalent:
(i) k cops have a winning strategy in GΓG0 .
(ii) k cops have a winning strategy in GΓG1 .
(iii) k cops have a winning strategy in GΓG2 .
(iv) k cops have a winning strategy in GΓG3 .
Proof. From the definitions provided in Example 5.2.1, it follows easily that a strategy
for the searchers in GΓG3 is a strategy in GΓG2 and also a strategy in GΓG1 ; a strategy for
the searchers in GΓG2 is a strategy in GΓG0 ; and a strategy in GΓG1 is also a strategy in
GΓG0 . Thus (iv)⇒(iii)⇒(i) and (iv)⇒(ii)⇒(i). We now show that (i)⇒(iv).
Suppose k cops have a winning strategy σ in GΓG0 . Let Γ0 (G) = (Lc , Lr , A),
and Γ3 (G) = (L0c , L0r , A0 ). Note that by the definition of Γ3 , L0c = Lc , L0r = Lr ,
V0 (A) = V0 (A0 ) and V1 (A) ⊇ V1 (A0 ). We show how to define a strategy σ 0 for k cops
such that for all (X, R) ∈ V0 (A0 ), |σ 0 (X, R) 4 X| ≤ 1. The idea is that we replace
each move of σ which involves moving more than one cop with a sequence of moves:
removing one cop at a time from X until cops remain on X ∩σ(X, R), and then adding
cops one at a time until they occupy σ(X, R). More formally, let Σ = V0 (A). We
define a history-dependent strategy σ 0 as follows. Let σ 0 (, X0 , R0 ) = (X0 , R0 ), ∅
where (X0 , R0 ) = vI (A). Now suppose w ∈ Σ∗ , w 6= , and the last symbol of w is
(X, R) ∈ V0 (A). Define σ 0 (w, X 0 , R0 ) as follows. If X ∩ σ(X, R) ⊂ X 0 ⊆ X, let
X 00 = X 0 \{v} for some v ∈ X 0 \σ(X, R), and define σ 0 (w, X 0 , R0 ) := (X, R), X 00 .
Otherwise, if X ∩ σ(X, R) ⊆ X 0 ⊂ σ(X, R), let X 00 = X 0 ∪ {v} for some v ∈
σ(X, R) \ X 0 , and define σ 0 (w, X 0 , R0) := (X, R), X 00 . Finally, if X 0 = σ(X, R)
define σ 0 (w, X 0 , R0 ) = (X 0 , R0 ), X 0 . Clearly σ 0 is a strategy for at most k cops
which involves placing or removing at most one cop at each step. We now show that it
is a winning strategy.
Let π = (X00 , R00 )(X10 , R10 ) · · · be a search consistent with σ 0 . Let w0 ∈ Σ∗ ∪Σω be
the history consistent with π, and let w be the word obtained by replacing repeated symbols in w0 with single occurrences. We observe that these repetitions arise where we
have replaced a single multiple-cop move with a finite sequence of single-cop moves
so w is infinite if, and only if, w0 is infinite. We also observe that by the definition of
σ 0 , w is a subsequence of π. We make the following claim:
Claim. The search defined by w is a search consistent with σ.
Proof of claim. Let w = (X1 , R1 )(X2 , R2 ) · · · . From the definition of σ 0 we have
Xi+1 = σ(Xi , Ri ) for all i ≥ 1, so it suffices to show that for all i ≥ 1 there is an
edge in A from (Xi , Xi+1 , Ri ) to (Xi+1 , Ri+1 ). That is, each possible set of locations
for the robber available after the sequence of single-cop moves is available after a
0
0
single multiple-cop move. Let m and n be such that (Xi , Ri ) = (Xm
, Rmax
) and
88
CHAPTER 5. GRAPH SEARCHING GAMES
(Xi+1 , Ri+1 ) = (Xn0 , Rn0 ) and let q be such that m ≤ q ≤ n and Xq0 = Xi ∩ Xi+1 .
0
We prove by induction that for all j, with m ≤ j ≤ n, Rj0 ∪ Rmax
is contained in a
0
connected component of G \ (Xm
∩ Xj0 ). Clearly this is true for j = m. Now suppose
0
0
for some j ≥ m, Rj0 ∪ Rmax
is contained in a connected component of G \ (Xm
∩ Xj0 ),
0
and consider Rj+1
. By the definition of the cops and visible robber game, Rj0 ∪ Rj+1
is contained in a connected component of G \ (Xj0 ∩ Xj+1 ). We consider the following
two cases. If j < q, then Xj+1 ⊆ Xj ⊆ Xm and Rj ⊇ Rmax . Thus the connected
0
component R of G \ Xj+1
which contains Rmax is the only component contained in
0
0
the same connected component of G \ (Xj+1
∩ Xj0 ) as Rj . Thus Rj+1
= R. Since
0
0
0
0
R ∪ Rmax = R is a connected component of G \ Xj+1 = G \ (Xj+1
∪ Xm
), our
0
hypothesis holds for j + 1. Now suppose j ≥ q. Then Xm ∩ Xj = Xi ∩ Xi+1 , and
0
0
0
Xj+1
⊇ Xj0 . Thus if Rj+1
is in the same connected component of G \ (Xj0 ∩ Xj+1
)=
0
0
0
0
0
G \ Xj as Rj , it follows that Rj ⊇ Rj+1 . By the inductive hypothesis, Rj is in the
0
0
0
same connected component of G \ (Xm
∩ Xj0 ) as Rmax
. But as G \ (Xm
∩ Xj0 ) =
0
0
0
G \ (Xi ∩ Xi+1 ) = G \ (Xm ∩ Xj+1 ), it follows that Rj+1 is in the same connected
0
0
0
component of G \ (Xm
∩ Xj+1
) as Rmax
. This completes the inductive step and the
proof of the claim.
a
Next we observe that as there is always a move available to the cops, π is winning
for the robber if, and only if, it is infinite. But this is the case if, and only if, w is
infinite. As σ is a winning strategy, there are no infinite searches consistent with σ,
thus π must be finite and therefore winning for the searchers.
u
t
Our final observation regarding the cops and visible robber game and the number
of cops required to capture the robber is a straightforward result which relates the game
and the resource measure with the parameterized class of games we also introduced in
Example 5.2.1.
Lemma 5.40. Let G be an undirected graph. The cops have a winning strategy in the
k-cops and visible robber game if, and only if, k cops have a winning strategy in the
cops and visible robber game.
Proof. Clearly a winning strategy σ for k cops in the cops and visible robber game is
a winning strategy for the cops in the k-cops and robber game: since |σ(X, R)| ≤ k
for all positions (X, R) in the cops and visible robber game, it follows that σ(X, R) ∈
[V (G)]≤k for all positions (X, R) in the k-cops and visible robber game.
For the converse, let σ be a winning strategy for the cops in the k-cops and robber
game. Let us extend σ to a strategy in the cops and visible robber game by defining
σ(X, R) = ∅ for all X ⊆ V (G) with |X| > k. Then, since |σ(X, R)| ≤ k for all
positions (X, R), σ is a strategy for k cops. Since any search in the cops and visible
robber game consistent with σ is also a search in the k-cops and visible robber game
consistent with σ, it follows that σ is a winning strategy in the cops and visible robber
game.
u
t
Remark. This example shows that with the resource measure ϕmax we can view resource bounded strategies as winning strategies in a parameterized family of graph
searching games. As such games are simple, if either the fugitive or the searchers have
5.3. COMPLEXITY MEASURES
89
a winning strategy, then they have a memoryless winning strategy. This justifies our
use of positional strategies in subsequent chapters.
Theorem 5.37 motivates the nomenclature used for Theorem 4.7: a haven is, as the
name suggests, a characterization of a winning strategy for the robber. Carrying this
reasoning to the definition of haven used in [JRST01], we see that Theorem 4.11 can
be restated as the following characterization of directed tree-width in terms of graph
searching games. We recall the strongly connected visible robber game defined in
Example 5.2.1.
Lemma 5.41. Let G be a digraph. Either G has directed tree-width ≤ 3k + 1 or k cops
do not have a winning strategy in the strong visible robber game on G.
5.3.2 Example: Cops and invisible robber
We now consider the resource measure ϕmax applied to the cops and invisible robber
game. Kirousis and Papadimitriou [KP86] showed that the number of cops required
to capture the robber in this game is equivalent to one more than the path-width of the
graph.
Theorem 5.42 ([KP86]). Let G be an undirected graph. The following are equivalent:
1. G has path-width ≤ k − 1.
2. k cops have a cop-monotone winning strategy in the cops and invisible robber
game.
3. k cops have a robber-monotone winning strategy in the cops and invisible robber
game.
4. k cops have a winning strategy in the cops and invisible robber game.
Together with Theorem 5.37, this theorem shows how we can view the relationship
between path-width and tree-width via graph searching games. As an example of the
consequence of this, Fomin, Fraigniaud and Nisse [FFN05] considered a parameterized family of cops and robber games where the robber is invisible, but the cops are
allowed q queries of the location of the robber during a search. The resulting family of
measures corresponding to the number of cops required in each game gives a parameterization which lies between path-width (q = 0) and tree-width (q = ∞). Because
such parameterized measures can be seen as a generalization of both path-width and
tree-width, they are particularly useful for investigating the structural complexity of
graphs.
5.3.3 Example: Cops and inert robber
We again consider the ϕmax resource measure, but this time with the cops and inert
robber game. Dendris, Kirousis and Thilikos [DKT97] showed that the number of
cops required to capture an invisible, inert robber is another measure equivalent to one
more than tree-width.
CHAPTER 5. GRAPH SEARCHING GAMES
90
Theorem 5.43 ([DKT97]). Let G be an undirected graph. The following are equivalent:
1. G has tree-width ≤ k − 1.
2. k cops have a robber-monotone winning strategy in the cops and inert robber
game.
3. k cops have a winning strategy in the cops and inert robber game.
Combining this with Theorem 5.37, we see that the number of cops required to
capture a robber in the cops and visible robber game is equal to the number of cops
required to capture a robber in the cops and inert robber game. In Chapter 7, where we
consider the generalization of the cops and inert robber game to directed graphs, we
show that this is not the case for the generalizations of the games to digraphs.
Dendris et al. also showed that the cop-monotone version of the cops and inert
robber game may require more cops than the robber-monotone version. In Chapter 7,
we show that the number of cops required in the cop-monotone version of the natural
extension of this game to directed graphs is equivalent to the extension of path-width
to digraphs.
5.3.4 Example: Other resource measures
We now consider some graph parameters which can be characterized by the invisible
and inert robber games, but with other resource measures. In [FG00], Fomin and Golovach considered the following resource measure which intuitively represents the “cost”
of a search.
Definition 5.44 (ϕcost ). The resource measure ϕcost is defined as follows. If π =
X1 X2 · · · is a sequence of finite sets, then
X
ϕcost (π) =
|Xi |.
i≥1
In [FG00] it was shown that the minimum cost of a search in a cops and invisible
robber game on a graph G is equivalent to the profile of G: the minimal number of
edges of an interval supergraph of G. In [FHT04] it was shown that the minimum cost
of a search in a cops and inert robber game on G is equivalent to the fill-in of G: the
minimum number of edges which need to be added to make G chordal. Summarizing
these results in our framework:
Theorem 5.45 ([FG00, FHT04]). Let Γ0 be the cops and invisible robber game type
defined in Definition 5.27 and let Γ1 be the cops and inert robber game type defined in
Definition 5.30. Let G be an undirected graph. Then
1. The profile of G is equal to w(Γ0 ,ϕcost ) (G).
2. The fill-in of G is equal to w(Γ1 ,ϕcost ) (G).
5.3. COMPLEXITY MEASURES
91
In [RS82] Rosenberg and Sudborough considered a pebbling game which Fomin et
al. [FHT04] observed can be seen as a version of the cops and invisible robber game.
Rosenberg and Sudborough showed that minimizing the resource measure defined by
the maximum life-time of a pebble on the graph is equivalent to finding the bandwidth
of the graph: the minimum, over all linear layouts of the vertices of the graph, of the
maximum distance between any pair of adjacent vertices. Fomin et al. [FHT04] viewed
this resource measure in the setting of graph searching games, to define the following
measure which indicates the “occupation time” of a search.
Definition 5.46 (ϕot ). Let π = X1 X2 · · · be a sequence of finite subsets of a set V . For
each i ≥ 1 let χi : V → {0, 1} be the characteristic function of Xi , so that χi (v) = 1
if, and only if, v ∈ Xi . Then ϕot is defined as follows:
X
χi (v).
ϕot (π) = max
v∈V
i≥1
Remark. In order for this measure to be non-trivial, we assume that we are working
with version of the cops and robber game where at most one cop is moved at a time.
The result of Rosenberg and Sudborough can then be summarized thus:
Theorem 5.47 ([RS82]). Let Γ be the cops and invisible robber game type defined in
Definition 5.27 where at most one cop is moved at a time, and let G be an undirected
graph. Then the bandwidth of G is equal to w(Γ,ϕot ) (G).
Fomin et al. [FHT04] used Theorem 5.47 to generate a generalization of bandwidth,
called treespan, by considering the resource measure ϕot on the cops and inert robber
game.
Theorem 5.48 ([FHT04]). Let Γ be the cops and inert robber game type defined in
Definition 5.30 where at most one cop is moved at a time, and let G be an undirected
graph. Then the treespan of G is equal to w(Γ,ϕot ) (G).
5.3.5 Monotonicity
Theorems 5.37, 5.42 and 5.43 all indicate an interesting property of some of the graph
searching games we have considered: the restriction imposed by bounding the resources supercedes the restriction imposed by monotonicity. This provides an explanation as to why measures like tree-width are good complexity measures from a practical
and structural perspective: winning strategies which are not necessarily monotone indicate the existence of various structural properties such as havens or brambles (as we
see in Chapter 8); on the other hand, monotone winning strategies are very useful algorithmically. As we saw with Lemma 5.11, monotone strategies can be represented
as restrictions on the arena, so it is often easier to compute monotone winning strategies. Furthermore, as we see in the next few chapters, monotone strategies often lend
themselves to decompositions with properties that make them very useful for practical
purposes. Thus it is important to identify games where monotonicity is not too great a
CHAPTER 5. GRAPH SEARCHING GAMES
92
restriction, as these games will provide measures that are good indicators of algorithmic and structural complexity. This leads to the question, “For which graph searching
game types and resource measures is monotonicity sufficient?” More precisely,
Open problem 5.49. For which graph searching game types Γ and resource measures
ϕ does (Γ, ϕ)-width give a bound on fugitive-monotone or searcher-monotone (G, ϕ)width?
Remark. Allowing approximate equivalence gives some flexibility in the above question: while it may not be the case that a winning strategy implies the existence of a
monotone winning strategy with the same resource bounds, it might still be possible
that the resource requirements for a monotone strategy can be deduced from those of a
winning strategy.
5.4 Robustness results
We now use the framework we have developed to show that the complexity measures
we have defined are well-behaved under some simple graph operations, thus indicating
their significance as a robust measure of graph complexity. In particular we show that,
under some reasonable assumptions, the width measure defined by a graph searching
game and a resource measure does not increase under the simplification operation of
taking subgraphs. We also show that the complexity measure we have defined can be
determined from the connected components of the graph. Finally, we consider the cops
and robber game. We show that the restriction of having the searchers and the fugitive
located on vertices enables us to show that the width measure defined by the number
of cops required in a cops and robber game suitably increases under a graph operation
which can be seen as a uniform complication, namely graph composition.
For convenience, we only consider width measures defined by order-preserving
resource measures. Thus for each of the following results, there is a dual result obtained
by replacing order-preserving with order-reversing, monotone with anti-monotone, and
≤ with ≥.
5.4.1 Subgraphs
In Definition 5.21 we introduced a restriction on graph searching game types, respecting restriction, which asserted that searching strategies in a graph G can be restricted to
be searching strategies in subgraphs of G. It turns out that imposing this restriction on
the graph searching game type and the monotonicity restriction on the resource measure is sufficient to show that graph searching width is well-behaved with respect to
subgraphs.
Theorem 5.50. Let Γ be a graph searching game type which respects restriction. Let
ϕ be a monotone, order-preserving resource measure. For any two graphs G, G 0 such
that G 0 is a subgraph of G:
w(Γ,ϕ) (G 0 ) ≤ w(Γ,ϕ) (G).
5.4. ROBUSTNESS RESULTS
93
Proof. Let GΓG and GΓG 0 be the graph searching games on G and G 0 defined by Γ(G)
and Γ(G 0 ) respectively. Since Γ respects restriction, it follows from Lemma 5.23 that
(⊃
· ,⊃
· ) is a searching simulation from GΓG to GΓG 0 . Let σ be a winning searcher strategy
in GΓG such that for any search π consistent with σ, ϕ(π) ≤ w(Γ,ϕ) (G). Let σ 0 be a
searching strategy in GΓG 0 (⊃
· ,⊃
· )-simulated by σ. It follows from Lemma 5.16 that
σ 0 is a winning strategy for the searchers. Furthermore, by the definition of σ 0 , for
any search π 0 = (X00 , R00 )(X10 , R10 ) · · · consistent with σ 0 there exists a search π =
(X0 , R0 )(X1 , R1 ) · · · consistent with σ such that Xi ⊃
· Xi0 for all i. Thus, Xi0 =
0
Xi ∩ Elts(G ) ⊆ Xi . Since ϕ is monotone, it follows that ϕ(π 0 ) ≤ ϕ(π) ≤ w(Γ,ϕ) (G),
and this holds for any search π 0 . Thus, from the definition of w(Γ,ϕ) (G 0 ), we have
w(Γ,ϕ) (G 0 ) ≤ w(Γ,ϕ) (G) as required.
u
t
In Lemma 5.18 we observed properties sufficient for a simulation to respect fugitive
and searcher-monotonicity. We now show that ⊃
· satisfies these properties, implying
that Theorem 5.50 can be extended to fugitive-monotone and searcher-monotone width.
0
Lemma 5.51. Let G and G 0 be graphs with G a subgraph of G 0 . The relation ⊃
· GG is
monotone and ∩-compatible.
0
0
Proof. Take X 0 , Y 0 ⊆ Elts(G 0 ) and X, Y ⊆ Elts(G) such that X 0 ⊃
· GG X and Y 0 ⊃
· GG
0
0
Y . From the definition of ⊃
· , it follows that X = X ∩ Elts(G) and Y = Y ∩ Elts(G).
0
Thus, if X 0 ⊆ Y 0 , X = X 0 ∩ Elts(G) ⊆ Y 0 ∩ Elts(G) = Y , so ⊃
· GG is monotone.
Furthermore, (X 0 ∩ Y 0 ) ∩ Elts(G) = (X 0 ∩ Elts(G)) ∩ (Y 0 ∩ Elts(G)) = X ∩ Y , so
0
0
(X 0 ∩ Y 0 ) ⊃
· GG (X ∩ Y ), and therefore ⊃
· GG is ∩-compatible.
u
t
Corollary 5.52. Let Γ be a graph searching game type which respects restriction. Let
ϕ be a monotone, order-preserving resource measure. For any two graphs G, G 0 such
that G 0 is a subgraph of G:
1. The fugitive-monotone (Γ, ϕ)-width of G is at most the fugitive-monotone (Γ, ϕ)width of G 0 , and
2. The searcher-monotone (Γ, ϕ)-width of G is at most the searcher-monotone (Γ, ϕ)width of G 0 .
5.4.2 Connected components
We now show how the widths of the connected components of a graph can be used to
compute the width of the graph. First we need to introduce a notion which is dual to
restriction respecting.
Definition 5.53 (Reflects restriction). Let Γ be a graph searching game type. We say
Γ reflects restriction if for any graphs G and G 0 such that G is a subgraph of G 0 , Γ(G) =
(Ls , Lf , A), and Γ(G 0 ) = (L0s , L0f , A0 ), then
0
• If Rmax is the ⊆-maximal element of Lf , and Rmax
is the ⊆-maximal element
0
0
of Lf , then Rmax = Rmax ∩ Elts(G).
94
CHAPTER 5. GRAPH SEARCHING GAMES
• If there is an edge from (Y, S) to (Y, Y 0 , S) in E(A) then for all (X, R) ∈
V0 (A0 ) and (X, X 0 , R) ∈ V1 (A0 ) such that Y = X ∩ Elts(G), S = R ∩ Elts(G)
and Y 0 = X 0 ∩ Elts(G), there is an edge in E(A0 ) from (X, R) to (X, X 0 , R),
and
• If there is an edge from (X, X 0 , R) to (X 0 , R0 ) in E(A0 ) and (Y, Y 0 , S) ∈ V (A)
for Y = X ∩ Elts(G),Y 0 = X 0 ∩ Elts(G) and S = R ∩ Elts(G), then either
R0 ∩ Elts(G) = ∅ or there is an edge from (Y, Y 0 , S) to (Y 0 , R0 ∩ Elts(G)) in
E(A).
Just as respecting restriction can be viewed as ⊃
· -closure, it would appear that restriction reflection should also be equivalent to R-closure for some quasi-simulation
family R similar to ⊃
· . However, the last condition in the definition is problematic for
the game simulation: the fugitive may be able to move in the larger graph (R0 6= ∅),
but because R0 ∩ Elts(G) = ∅, there is no response on the smaller graph. Nevertheless,
we are able to obtain a result, similar to Lemma 5.13, sufficient for our purposes.
Lemma 5.54. Let Γ be a graph searching game type which reflects restriction and let
G and G 0 be graphs such that G is a subgraph of G 0 . Let Γ(G) = (Ls , Lf , A), Γ(G 0 ) =
(L0s , L0f , A0 ), and take (X00 , R00 ) ∈ V (A0 ) such that X00 ∩Elts(G) = ∅ and R00 ∩Elts(G)
is either ∅ or the ⊆-maximal element of Lf . If σ is a winning strategy for the searchers
in GΓG , then there exists a strategy σ
e for the searchers on GΓG 0 such that any search
0
0
from (X0 , R0 ) consistent with σ
e can be extended to a search (X00 , R00 )(X10 , R10 ) · · ·
consistent with σ
e so that there exists n ≥ 0 with Rn0 ∩ Elts(G) = ∅, and for all i,
0
1 ≤ i ≤ n, Xi = σ(Xi−1 , Ri−1 ) for some (Xi−1 , Ri−1 ) ∈ V0 (A).
Proof. For (X 0 , R0 ) ∈ V (A0 ) with (X, R) ∈ V (A) where X = X 0 ∩ Elts(G) and
R = R0 ∩ Elts(G), define σ
e (X 0 , R0 ) := σ(X, R). From the second condition of
restriction reflection, this is a well-defined (partial) strategy: (X 0 , σ(X 0 , R0 ), R0 ) is
a successor of (X 0 , R0 ). We now show that σ
e is sufficiently defined to satisfy the
requirements of the lemma.
Let π 0 = (X00 , R00 )(X10 , R10 ) · · · (Xn0 , Rn0 ) be a search from (X00 , R00 ) consistent
with σ
e. For i ≥ 0, let Xi = Xi0 ∩ Elts(G) and Ri = Ri0 ∩ Elts(G). By the definition
of σ
e , Xi0 = σ(Xi−1 , Ri−1 ) for all i such that Ri−1 6= ∅. Thus if we take n to be the
minimum index such that Rn = ∅, we are done. So suppose there is no n such that
Rn = ∅. We claim:
Claim. π = (X0 , R0 )(X1 , R1 ) · · · is a search from vI (A) consistent with σ.
Proof of claim. We prove this by induction on i, the length of π consistent with σ.
From the definition of (X00 , R00 ), and since R00 ∩ Elts(G) 6= ∅, (X0 , R0 ) = vI (A), so
the claim is true for i = 0. Now suppose (X0 , R0 ) · · · (Xi , Ri ) is consistent with σ.
0
0
0
, Ri+1
)
From the definition of σ 0 , Xi+1 = Xi+1
= σ(Xi , Ri ). As (X00 , R00 ) · · · (Xi+1
0
is consistent with σ
e, and Ri+1 ∩ Elts(G) 6= ∅, it follows that there is an edge in
0
0
0
E(A0 ) from (Xi0 , Xi+1
, Ri0 ) to (Xi+1
, Ri+1
). Thus, from the third condition of restriction reflection, there is an edge from (Xi , Xi+1 , Ri ) to (Xi+1 , Ri+1 ). Therefore,
(X0 , R0 ) · · · (Xi+1 , Ri+1 ) is consistent with σ as Xi+1 = σ(Xi , Ri ) and there is an
edge from (Xi , Xi+1 , Ri ) to (Xi+1 , Ri+1 ).
a
5.4. ROBUSTNESS RESULTS
95
Now, since σ is a winning strategy for the searchers, every search from vI (A)
consistent with σ can be extended to a complete search. However, Ri 6= ∅ for all i ≥ 0,
so π cannot be extended to a complete search. Thus there exists n such that Rn = ∅,
contradicting the assumption that there is no such n.
u
t
We also need to assume that our graph searching games satisfy the following property: if the searchers have a winning strategy from (X, R) then the searchers can play
the same strategy and win from (X, S) for any S ⊆ R. To be more precise, we require
the graph searching game type to be (id, ⊇)-closed where id is the quasi-simulation
family which assigns to each pair of graphs (G, G 0 ) with G = G 0 the identity relation,
and ⊇ is the quasi-simulation family which assigns to each pair of graphs (G, G 0 ) with
G = G 0 the superset relation. Given such a graph searching game type, we can apply
Lemma 5.13 to obtain the following:
Lemma 5.55. Let Γ be a graph searching type which is (id, ⊇)-closed, and let G be a
graph with Γ(G) = (Ls , Lf , A). For any (X1 , R1 ), (X10 , R10 ) ∈ V (A) with X1 = X10
and R1 ⊇ R10 and any strategy σ for the searchers on GΓG , there exists a strategy for the
searchers σ 0 on GΓG such that for every search (X10 , R10 )(X20 , R20 ) · · · consistent with
σ 0 , there exists a search (X1 , R1 )(X2 , R2 ) · · · consistent with σ with Xi = Xi0 and
Ri ⊇ Ri0 for all i.
To compute the width of a graph from the widths of its connected components, we
need to be able to combine the widths of the components. To do this we require some
sort of operation, ⊕, on ω which reflects how our resource measure is computed. For
example, if we are interested in the number of searchers required to capture a fugitive,
then the function max is the combining operation we are interested in, the number of
searchers required in the whole graph is at most the maximum number of any of its
components. In fact, we can use any operation ⊕ for which our resource measure is
“well-behaved”, in the following sense:
Definition 5.56 (⊕-morphism). Let ϕ be a resource measure and ⊕ : ω × ω → ω
an operation on ω. We say ϕ is a ⊕-morphism if ϕ(π · π 0 ) = ϕ(π) ⊕ ϕ(π 0 ) for all
sequences π and π 0 .
Remark. We note that if ϕ is a ⊕-morphism, then (on the image of ϕ) the operation ⊕
is uniquely defined. That is, for any resource measure ϕ, there is at most one possible
operation ⊕ such that ϕ is a ⊕-morphism. However, we also observe that given any
monoid structure (id, ⊕) on ω and a function f from finite sets to ω, we can define a
⊕-morphism ϕ⊕ as follows:
ϕ⊕ () =
ϕ⊕ (X1 · · · Xn ) =
ϕ⊕ (π) =
id,
f (X1 ) ⊕ · · · ⊕ f (Xn ), and
ω if π is infinite.
We also note that if ϕ is a ⊕-morphism, then, due to the associativity of concatenation, ⊕ is necessarily associative. That is, if a = ϕ(πa ), b = ϕ(πb ), and c = ϕ(πc ),
CHAPTER 5. GRAPH SEARCHING GAMES
96
then we have:
(a ⊕ b) ⊕ c
=
=
=
=
ϕ(πa ) ⊕ ϕ(πb ) ⊕ ϕ(πc )
ϕ (πa · πb ) · πc
ϕ πa · (πb · πc )
ϕ(πa ) ⊕ ϕ(πb ) ⊕ ϕ(πc ) = a ⊕ (b ⊕ c).
Our next observation is that if we combine the restrictions we have just introduced,
then the combination of the widths of the components of a graph provides an upper
bound on the width of the graph.
Lemma 5.57. Let Γ be a graph searching game type which reflects restriction and
is (id, ⊇)-closed. Let ϕ be an order-preserving ⊕-morphism. If G is a graph with
(weakly) connected components G1 , G2 , . . . , Gn , then:
w(Γ,ϕ) (G) ≤
n
M
w(Γ,ϕ) (Gj ).
j=1
Proof. Let Γ(G) = (Ls , Lf , A) and for 1 ≤ j ≤ n, let Γ(Gj ) = (Ljs , Ljf , Aj ). For
convenience, for each set X ⊆ Elts(G), let X j = X ∩ Elts(Gj ). Note that since
j
Γ reflects restriction, if Rmax is the ⊆-maximal element of Lf , then Rmax
is the ⊆j
maximal element of Lf . For each j, 1 ≤ j ≤ n, let σj be a winning strategy for the
searchers such that for every search πj in GΓGj consistent with σj , ϕ(πj ) ≤ w(Γ,ϕ) (Gj ).
The idea is that the strategy defined by playing each of the strategies
Lnσj sequentially
is a winning strategy which has a resource requirement of at most j=1 w(Γ,ϕ) (Gj ).
Before we formally define the strategy, we make the following observation.
Claim. Let (X1 , R1 )(X2 , R2 ) · · · be a search in GΓG . For any j, 1 ≤ j ≤ n, if there
exists n ≥ 0 such that Rnj = ∅, then Rij = ∅ for all i ≥ n.
Proof of claim. Fix j, and suppose n is such that Rnj = ∅. Suppose there exists i > n
such that Rij 6= ∅. Let k be the minimal index such that Rkj 6= ∅, and take r0 ∈ Rkj .
From the definition of a graph searching game, there exists r ∈ Rk−1 such that r
and r0 are in the same (weakly) connected component of G \ (Xk−1 ∩ Xk ). Thus,
j
as r0 ∈ Elts(Gj ), it follows that r ∈ Elts(Gj ). Thus r ∈ Rk−1
, contradicting the
j
minimality of k. Therefore Ri = ∅ for all i ≥ n.
a
We define σ inductively as follows. If G has one connected component, let σ = σ1 .
Clearly σ is a winning strategy on G, and for any search π consistent
with σ we have
Sn
ϕ(π) ≤ w(Γ,ϕ) (G1 ). Now consider the subgraph G 0 = j=2 Gj . Let Γ(G 0 ) =
(L0s , L0f , A0 ). Suppose there exists a winning strategy σ0 on GΓG 0 such that for any
Ln
search π consistent with σ0 we have ϕ(π) ≤
j=2 w(Γ,ϕ) (Gj ). Using the notation
from Lemma 5.54, let σ
f0 be the strategy on GΓG defined by σ0 , and let σ
f1 be the strategy on GΓG defined by σ1 . The strategy σ is as follows: from (∅, Rmax ), play σ
f1 until a
position (X, R) is reached where R ∩ V (G1 ) = ∅. That is, until R0 ∩ V (G 0 ) = ∅, let
5.4. ROBUSTNESS RESULTS
97
σ(X 0 , R0 ) = σ
f1 (X 0 , R0 ). From Lemma 5.54, we have X ⊆ V (G1 ), so X ∩V (G 0 ) = ∅,
0
0
and since R ⊆ V (G 0 ), R ∩ V (G 0 ) ⊆ Rmax
where Rmax
is the ⊆-maximal element of
0
L0f . Thus (X, Rmax
) is (id, ⊇)-related to (X, R). Since X ∩ V (G 0 ) = ∅, it follows
0
that f
σ0 (X, Rmax
) is defined. Let σ00 be a (id, ⊇)-simulated strategy of σ
f0 , which, from
Lemma , plays from (X, R) when σ
f0 plays from (X, Rmax ). For all subsequent positions (X 0 , R0 ) reached, including (X, R), define σ(X 0 , R0 ) = σ00 (X 0 , R0 ). From the
earlier claim, as R0 ∩ V (G1 ) = ∅, it follows from the definition of simulated strategies
that σ is well-defined for all subsequent positions. As σ1 and σ 0 are winning strategies,
it also follows that σ is a winning strategy.
Let us now consider the resources required by σ. Let π = (X0 , R0 )(X1 , R1 ) · · ·
be a search consistent with σ. From the definition of σ, it follows that π = π1 ·π 0 where
π1 is a search consistent with σ
f1 and π 0 is a search consistent with σ00 . Therefore, from
Lemmas 5.54 and 5.55, it follows that the sequence π = X0 X1 · · · is equal to π1 · π 0
where π1 is the sequence of first components of a search consistent with σ1 and π 0 is
the sequence of first components of a search consistent with σ 0 . Thus
ϕ(π) = ϕ(π1 · π 0 ) = ϕ(π1 ) ⊕ ϕ(π 0 )
n
n
M
M
w(Γ,ϕ) (Gj ).
w(Γ,ϕ) (Gj ) =
≤ w(Γ,ϕ) (G1 ) ⊕
j=1
j=2
As this holds L
for any play consistent with σ, and σ is a winning strategy, it follows that
n
u
t
w(Γ,ϕ) (G) ≤ j=1 w(Γ,ϕ) (Gj ).
If we impose some further restrictions on the operation ⊕, and suitable restrictions
on Γ and ϕ, we can use Theorem 5.50 to obtain equality in the above result.
Definition 5.58. Let ⊕ : ω × ω → ω be an operation on ω. We say ⊕ is monotone if
for all a, b, c, d ∈ ω with a ≤ b and c ≤ d, a ⊕ c ≤ b ⊕ d. We say ⊕ is deflationary if
for all a ∈ ω, a ≥ a ⊕ a.
Theorem 5.59. Let Γ be a graph searching game type which respects and reflects
restriction and is (id, ⊇)-closed. Let ⊕ : ω × ω → ω be an associative, monotone, and
deflationary operation on ω. Let ϕ be a monotone, order-preserving ⊕-morphism. If G
is a graph and G1 , G2 , . . . , Gn are the (weakly) connected components of G, then,
w(Γ,ϕ) (G) =
n
M
w(Γ,ϕ) (Gj ).
j=1
L
Proof. From Lemma 5.57, we have w(Γ,ϕ) (G) ≤ nj=1 w(Γ,ϕ) (Gj ). For the reverse inequality, we observe that as Gj is a subgraph of G for all j, we have from Theorem 5.50,
w(Γ,ϕ) (Gj ) ≤ w(Γ,ϕ) (G) for all j. Thus, as ⊕ is deflationary and monotone:
w(Γ,ϕ) (G) ≥
n
M
j=1
w(Γ,ϕ) (G) ≥
n
M
w(Γ,ϕ) (Gj ).
j=1
u
t
CHAPTER 5. GRAPH SEARCHING GAMES
98
5.4.3 Lexicographic product
We now consider the cops and robber game with the resource measure that indicates the
maximum number of cops used by a strategy, ϕmax . We show that, under some simple
assumptions, if we replace vertices in a graph with copies of a complete graph with n
vertices, the number of cops required to capture the robber increases by a factor of n.
We recall from Section 1.1.2 the definition of the lexicographic product. We now introduce some useful relations between a graph and its lexicographic factors. Although
these definitions are quite technical, later in the section we introduce some more intuitive properties which we show are sufficient to establish the robustness results we are
interested in.
Definition 5.60 (MH , DH and PH ). Let G and H be graphs and let G 0 = G • H. We
define MGH ⊆ P(V (G)) × P(V (G 0 )) and DGH , PGH ⊆ P(V (G 0 )) × P(V (G)) as follows.
If A ⊆ V (G) and B ⊆ V (G 0 ), then
• A MGH B if B = A × V (H),
• B DGH A if A = {u : (u, v) ∈ B for all v ∈ V (H)},
• B PGH A if A = {u : (u, v) ∈ B for some v ∈ V (H)}.
The following results follow immediately from Lemma 5.16 and provide an idea of
the results we are interested in.
Lemma 5.61. Let G and H be graphs and let G 0 = G •H. Let GΓG be a cops and robber
0
game on G and GΓG 0 be a cops and robber game on G 0 . If (MGH , MGH ) is a searching
0
simulation from GΓG to GΓG 0 and k cops have a winning strategy on GΓG , then k · |V (H)|
0
cops have a winning strategy on GΓG 0 .
Proof. Let σ be a winning strategy for the cops on GΓG which uses at most k cops. Let
0
σ 0 be a strategy for the cops on GΓG 0 (MGH , MGH )-simulated by σ. From Lemma 5.16,
σ 0 is a winning strategy for the cops. From the definition of MGH , for each position
0
(X 0 , R0 ) of GΓG 0 we have σ 0 (X 0 , R0 ) = σ(X, R) × V (H) for some position (X, R) of
GΓG . So |σ 0 (X 0 , R0 )| ≤ k · |V (H)|, and therefore σ 0 is a winning strategy for at most
k · |V (H)| cops.
u
t
Lemma 5.62. Let G and H be graphs and let G 0 = G •H. Let GΓG be a cops and robber
0
game on G and GΓG 0 be a cops and robber game on G 0 . If (DGH , PGH ) is a searching
0
simulation from GΓG 0 to GΓG and the robber can defeat k −1 cops on GΓG , then the robber
0
can defeat k · |V (H)| − 1 cops on GΓG 0 .
Proof. We consider the contrapositive: suppose k · |V (H)| − 1 cops have a winning
0
strategy σ 0 on GΓG 0 . We show that k − 1 cops have a winning strategy on GΓG . Let σ be
a strategy (DGH , PGH )-simulated by σ 0 . From Lemma 5.16, σ is a winning strategy for
the cops. Suppose |σ(X, R)| ≥ k for some position (X, R). From the definition of σ,
0
there exists a position (X 0 , R0 ) of GΓG 0 such that σ 0 (X 0 , R0 ) DGH σ(X, R). But then, as
|σ(X, R)| ≥ k, |σ 0 (X 0 , R0 )| ≥ k · |V (H)|, contradicting the assumption that σ 0 was a
strategy for k · |V (H)| − 1 cops. Thus σ 0 is a winning strategy for k − 1 cops.
u
t
5.4. ROBUSTNESS RESULTS
99
With these two results in mind, we introduce two quasi-simulation families which
we use to define the restriction on graph searching game types that we require for
games to be well-behaved under lexicographic product.
Definition 5.63 (Composition-expanding). Let M be the quasi-simulation family which
assigns to each pair of graphs (G, G 0 ), where G 0 = G • K for some complete graph K,
the pair of relations (MGK , MGK ). Let D be the quasi-simulation family which assigns
to each pair of graphs (G 0 , G), where G 0 = G •K for some complete graph K, the pair of
relations (DGK , PGK ). Let Γ be a cops and robber game type. We say Γ is compositionexpanding if it is M-closed and D-closed.
Using Lemmas 5.61 and 5.62, we obtain:
Theorem 5.64. Let Γ be a composition-expanding cops and robber game type. Let G
be a graph, and let Kn be the complete graph on n vertices. Then
n · w(Γ,ϕmax ) (G) = w(Γ,ϕmax ) (G • Kn ).
Proof. Let w(Γ,ϕmax ) (G) = k and w(Γ,ϕmax ) (G • Kn ) = m. From Lemma 5.61, we
have m ≤ n · k, so suppose m = n · k − r. But if r ≥ 1, then by Lemma 5.62,
w(Γ,ϕmax ) (G) ≤ k − 1. Thus r = 0 and the result follows.
u
t
To help identify cops and robber game types which are composition-expanding, we
now present an alternative characterization of composition-expanding, similar to the
definition of restriction respecting in Definition 5.21. Just as with Lemma 5.23, the
proof follows directly from the definitions, and is therefore omitted.
Lemma 5.65. Let Γ be a cops and robber game type such that for all graphs G and all
complete graphs K, where Γ(G) = (Lc , Lr , A), Γ(G • K) = (L0c , L0r , A0 ) and:
(I) If there is an edge in E(A) from (Y, S) to (Y, Y 0 , S) and (X, R) ∈ V0 (A0 ) for
X = Y × V (K) and R = S × V (K), then there is an edge in E(A0 ) from (X, R)
to (X, X 0 , R) where X 0 = Y 0 × V (K);
(II) If there is an edge in E(A0 ) from (X, R) ∈ V0 (A0 ) to (X, X 0 , R) and (Y, S) ∈
V0 (A) for Y = {u : (u, v) ∈ X for all v ∈ V (K)} and S = {u : (u, v) ∈
R for some v ∈ V (K)} , then there is an edge in E(A) from (Y, S) to (Y, Y 0 , S)
where Y 0 = {u : (u, v) ∈ X 0 for all v ∈ V (K)};
(III) If there is an edge in E(A0 ) from (X, X 0 , R) to (X 0 , R0 ) and (Y, Y 0 , S) ∈ V1 (A)
where X = Y × V (K), X 0 = Y 0 × V (K), and R = S × V (K); and then
R0 = S 0 × V (K) for some S 0 and there is an edge in E(A) from (Y, Y 0 , S) to
(Y 0 , S 0 )
(IV) If there is an edge in E(A) from (Y, Y 0 , S) to (Y 0 , S 0 ) and (X, X 0 , R) ∈ V1 (A0 )
where Y = {u : (u, v) ∈ X for all v ∈ V (K)}, Y 0 = {u : (u, v) ∈ X 0 for all v ∈
V (K)}, and S = {u : (u, v) ∈ R for some v ∈ V (K)}, then there is an edge
in E(A0 ) from (X, X 0 , R) to (X 0 , R0 ) for some R0 such that S 0 = {u : (u, v) ∈
R0 for some v ∈ V (K)},
100
CHAPTER 5. GRAPH SEARCHING GAMES
then Γ is composition-expanding.
We observed in Lemma 5.51 that the ⊃
· relation satisfied the necessary conditions
for (⊃
· ,⊃
· )-simulation to respect fugitive and searcher-monotonicity. We now show that
the relations M, D, and P also satisfy similar conditions implying that Theorem 5.64
holds for robber-monotone and cop-monotone width.
Lemma 5.66. Let G be a graph and K a complete graph.
1. The relation MGK is monotone and ∩-compatible.
2. The relation DGK is monotone and ∩-compatible.
3. The relation PGK is monotone.
Proof. 1: Take X, Y ⊆ V (G) and X 0 , Y 0 ⊆ V (G • K) such that X MGK X 0 and
Y MGK Y 0 . By the definition of MGK , it follows that X 0 = X × V (K) and Y 0 =
Y × V (K). So if X ⊆ Y , X 0 ⊆ Y 0 , and so MGK is monotone. Furthermore, since
(X ∩ Y ) × V (K) = (X × V (K)) ∩ (Y × V (K)), it follows that MGK is ∩-compatible.
2: Take X, Y ⊆ V (G) and X 0 , Y 0 ⊆ V (G • K) such that X 0 DGK X and Y 0 DGK
Y . By the definition of DGK , it follows that X = {u : (u, v) ∈ X 0 for all v ∈ V (K)}
and Y = {u : (u, v) ∈ Y 0 for all v ∈ V (K)}. Now if X 0 ⊆ Y 0 , it follows that
X = {u : (u, v) ∈ X 0 for all v ∈ V (K)} ⊆ {u : (u, v) ∈ Y 0 for all v ∈ V (K)} = Y .
Thus DGK is monotone. Furthermore, {u : (u, v) ∈ X 0 ∩ Y 0 for all v ∈ V (K)} = {u :
(u, v) ∈ X 0 for all v ∈ V (K)} ∩ {u : (u, v) ∈ Y 0 for all v ∈ V (K)}, so (X 0 ∩ Y 0 ) DGK
X ∩ Y , and hence DGK is ∩-compatible.
3: Take X, Y ⊆ V (G) and X 0 , Y 0 ⊆ V (G • K) such that X 0 PGK X and Y 0 PGK Y .
By the definition of PGK , it follows that X = {u : (u, v) ∈ X 0 for some v ∈ V (K)}
and Y = {u : (u, v) ∈ Y 0 for some v ∈ V (K)}. Now if X 0 ⊆ Y 0 , it follows that X =
{u : (u, v) ∈ X 0 for some v ∈ V (K)} ⊆ {u : (u, v) ∈ Y 0 for some v ∈ V (K)} = Y .
Thus PGK is monotone.
u
t
Corollary 5.67. Let Γ be a composition-expanding cops and robber game type. Let G
be a graph, and let Kn be the complete graph on n vertices. Then:
1. The robber-monotone (Γ, ϕmax )-width of G • Kn is n times the robber-monotone
(Γ, ϕmax )-width of G.
2. The cop-monotone (Γ, ϕmax )-width of G • Kn is n times the cop-monotone
(Γ, ϕmax )-width of G.
5.5 Complexity results
To conclude this chapter we consider the complexity of the problem of determining
the (Γ, ϕ)-width of a graph. More precisely, for a graph searching game type Γ and
an order-preserving resource measure ϕ, we are interested in the complexity of the
following problem:
5.5. COMPLEXITY RESULTS
101
(Γ, ϕ)-WIDTH
Instance: A graph G and k ∈ ω
Problem: Is the (Γ, ϕ)-width of G at most k?
Of course, the complexity of this problem is dependent on how difficult it is to
compute the arena of GΓG and the resource function ϕ. To have a sensible analysis, we
assume that we can compute these in amortized constant time, that is, we can compute
a path of length n in the arena, or the ϕ-value of a sequence of n sets in time O(n).
In practice computing edges of the arena and values of ϕ are more likely to require
time polynomial in the size of the graph, but as the bounds we obtain are generally
exponential in the size of the graph, this assumption is not going to significantly affect
the overall complexity.
From Definition 5.1, we know that a graph searching game GΓG defined by (Ls , Lf , A)
is a simple game, so it might appear at first that determining if the searchers have a winning strategy can be decided in time linear in the size of the arena, as per Theorem 2.60.
However, for an arbitrary resource measure ϕ, whether a vertex of the arena is winning
for the searchers in the resource-bounded game is dependent on the play to that vertex.
So it could be the case that for any strategy, all possible consistent plays have to be
checked to ensure the resource measure is bounded. Hence it may not be possible to do
better than to iterate through all possible strategies and all consistent searches, or equivalently, all possible plays in the arena. However, as we observed after Definition 5.36,
we need only consider plays that are simple paths in the arena, so this is at least decidable. Since every play can be characterized by a search, and a search is a sequence
of positions, there are at most O(|V0 (A)|!) plays that might have to be checked. Now
V0 (A) consists of pairs of subsets of Elts(G), thus |V0 (A)| = O(4|Elts(G)| ) = O(4||G|| ),
giving us the following bound:
Proposition 5.68. Let Γ be a graph searching game type and ϕ an order-preserving
resource measure. (Γ, ϕ)-WIDTH can be decided in time O(4n !).
We can do considerably better by considering specific resource measures, in particular the measure ϕmax . In Lemma 5.40, we saw how the existence of a resource
bounded winning strategy is equivalent to the existence of a winning strategy in a game
with a smaller arena: the parameterized game defined in Definition 5.26. We can use
Theorem 2.60 to decide if the cops have a winning strategy in this parameterized game
in linear time, and therefore determine if the cops have a resource bounded winning
strategy in the original game. More precisely,
Proposition 5.69. Let Γ be the cops and visible robber game type defined in Definition 5.24. Then (Γ, ϕmax )- WIDTH can be decided in time O(n2k+4 ).
Proof. Suppose G, an undirected graph, and k ∈ ω are given. Let Γ0 be the k-cops and
visible robber game type defined in Definition 5.26, and suppose Γ0 (G) = (Lc , Lr , A).
From Lemma 5.40, we have that k cops have a winning strategy in GΓG if, and only if,
0
the cops have a winning strategy in GΓG . From Theorem 2.60, we can determine if the
0
cops have a winning strategy in GΓG in time O(|E(A)|), so it suffices to find an upper
bound on |E(A)|. From the definition of the game, we observe that for each X, X 0 ∈
102
CHAPTER 5. GRAPH SEARCHING GAMES
Lc there are at most |V (G)| sets R such that (X, R) ∈ V0 (A) and (X, X 0 , R) ∈ V1 (A).
Therefore, from the definition of A we see that each element (X, X 0 , R) of V1 (A) has a
unique incoming edge (from (X, R)) and at most |V (G)| outgoing edges (to (X 0 , R0 )).
Thus the number of edges is at most (|V (G)| + 1)|V1 (A)|. From the definition of Lc ,
we have |Lc | ≤ |V (G)|k+1 , thus |V1 (A)| is at most |Lc ||Lc ||V (G)| ≤ |V (G)|2k+3 .
Therefore, the number of edges of A is bounded by O(|V (G)|2k+4 ), and the result
follows.
u
t
The parameterized class of games we defined in Definition 5.26 is easily extended
to other graph searching game types, so we can use a similar argument as above to
decide (Γ, ϕmax )- WIDTH more efficiently than Proposition 5.68. In the more general
case, we may not be able to bound the size of V1 (A) as efficiently, nor the number of
outgoing edges from elements of V1 (A). However, we observe that V1 (A) ⊆ Ls ×Ls ×
Lf , so |V1 (A)| ≤ ||G||k · ||G||k · 2||G||, and there are at most |Lf | ≤ 2||G|| outgoing edges
from any element of V1 (A). This gives us the following improvement for deciding
(Γ, ϕmax )- WIDTH:
Proposition 5.70. Let Γ be a graph searching game type. (Γ, ϕmax )- WIDTH can be
decided in time O(n2k+2 4n ).
We observe that all the algorithms we have so far considered are constructive: if
the algorithm returns a positive answer, then it is possible to extract a winning strategy
for the searchers.
We conclude the section by considering the complexity of determining the fugitivemonotone and searcher-monotone widths of a graph. As we observed following Lemma 5.11,
the restriction to fugitive-monotone strategies can be enforced by removing edges from
the arena. It therefore follows that the bounds we obtained for the general games are
applicable to the fugitive-monotone case.
Proposition 5.71. Let Γ be a graph searching game type.
(i) F UGITIVE - MONOTONE (Γ, ϕmax )- WIDTH can be decided in time O(n2k+2 4n ),
and
(ii) If Γ is the cops and visible robber game type defined in Definition 5.24. Then
F UGITIVE - MONOTONE (Γ, ϕmax )- WIDTH can be decided in time O(n2k+4 ).
Unfortunately, for searcher-monotone strategies the situation is not as straightforward. Indeed, just as with arbitrary resource measures, the algorithm of Theorem 2.60
cannot, in general, be used as the set of successors available from (X, R) is dependent
on the play to (X, R). Thus in the searcher-monotone case, we can in general do no
better than the bounds obtained for an arbitrary resource measure.
Proposition 5.72. Let Γ be a graph searching game type. S EARCHER - MONOTONE
(Γ, ϕmax )- WIDTH can be decided in time O(4n !).
Chapter 6
Digraph measures: DAG-width
In Chapter 4 we discussed the problem of finding a measure of complexity for digraphs.
We reviewed the definition of tree-width, arguably one of the most suitable measures of
complexity for undirected graphs, and we considered the problem of finding a suitable
generalization of tree-width for directed graphs. In Chapter 5 we introduced graph
searching games, a useful tool for developing robust measures of graph complexity,
and saw that several such games can be used to characterize tree-width. In this chapter we introduce a complexity measure for directed graphs which we argue is a more
natural generalization of tree-width than directed tree-width. We introduce a decomposition which, unlike arboreal decompositions, is defined in a similar manner to tree
decompositions. Just as tree decompositions are decompositions based on trees, our
decompositions are based on directed, acyclic graphs (DAGs), so we use the name
DAG-decompositions. And just as tree decompositions give rise to tree-width, DAGdecompositions give rise to a graph parameter which we call DAG-width.
We show that DAG-decompositions and DAG-width enjoy many properties similar
to tree decompositions and tree-width. For example, in Theorem 6.28, we show that we
may assume a DAG-decomposition satisfies certain conditions similar to those of nice
tree decompositions, introduced in [Bod97]. This normalized form is particularly useful for designing dynamic programming algorithms which run efficiently on classes of
directed graphs of bounded DAG-width. We see this in Section 6.3.3 when we present
such an algorithm for parity games. But perhaps the strongest point in favour of DAGwidth being a more natural generalization of tree-width is that it can be characterized
by a natural generalization of the cops and visible robber game, a graph searching game
which we saw in Chapter 5 characterizes tree-width. As the generalized game is particularly dependent on directed paths in the graph, this suggests that DAG-width is a good
indicator of the directed connectivity of a digraph, a notion we discussed in Chapter 4.
The game characterization of DAG-width also provides support for the argument
that DAG-width is a good measure of digraph complexity. For example, it is straightforward to show that DAG-width does not increase under the taking of subgraphs, and
that the DAG-width of a graph can be computed from the DAG-width of its strongly
connected components.
After we introduce DAG-width and its associated graph searching game, we con103
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CHAPTER 6. DAG-WIDTH
sider the algorithmic benefits of DAG-width. As a digraph measure, DAG-width lies
between tree-width and directed tree-width. That is, classes of graphs of bounded treewidth have bounded DAG-width and graphs of bounded DAG-width have bounded
directed tree-width. In particular this implies that algorithms which are efficient on
graphs of bounded directed tree-width are efficient on graphs of bounded DAG-width,
so in particular Theorem 4.13 applies also to graphs of bounded DAG-width. In this
chapter we extend this algorithmic result and show that parity games can be decided
in polynomial time on arenas of bounded DAG-width, something which is not currently known for graphs of bounded directed tree-width. We also show that DAGwidth, tree-width and directed tree-width are different measures by exhibiting a class
of digraphs with bounded DAG-width and unbounded tree-width and a class of digraphs with bounded directed tree-width and unbounded DAG-width. This suggests
that weak connectivity, directed connectivity and strong connectivity are three very
different properties of directed graphs.
The chapter is arranged as follows. In Section 6.1 we introduce the cops and visible robber game for directed graphs and we establish some results to help gain an
understanding of the game. We then define DAG-decompositions in Section 6.2, and
show the equivalence between DAG-width and the number of cops required to capture
the fugitive with a monotone strategy. In Section 6.3 we discuss some algorithmic aspects of DAG-width. We also prove the existence of a polynomial time algorithm for
solving parity games on arenas of bounded DAG-width, and in Section 6.4 we relate
DAG-width to other measures of graph connectivity, in particular tree-width, directed
tree-width and directed path-width.
6.1 Cops and visible robber game
We recall from Chapter 5 the cops and visible robber game from Example 5.2.1. In this
game a number of cops and a robber occupy vertices of an undirected graph and the
objective of the cops is to capture the robber. The cops move by removing some of their
number from the graph and announcing a set of vertices to be occupied. Following this,
the robber can move at great speed along paths in the graph to avoid capture, however
he is not permitted to pass through any cop which remains on the graph. The cops then
occupy the vertices that were announced, and if the robber is located on one of these
vertices then he is captured. The location of the robber in the graph is always known
to the cops. In Theorem 5.37 we saw that the minimum number of cops required to
capture a robber on an undirected graph is equal to one more than the tree-width of the
graph.
We now consider the natural extension of this game to directed graphs, where the
robber is constrained to move along directed cop-free paths. More precisely,
Definition 6.1 (Cops and visible robber game). Let G be a directed graph. The cops
and visible robber game on G is the cops and robber game defined by (Lc , Lr , A),
where
• Lc = P(V (G)) and Lr = P(V (G)) \ {∅},
6.1. COPS AND VISIBLE ROBBER GAME
105
• V0 (A) consists of (∅, V (G)) together with pairs (X, R) ∈ Lc × Lr such that
R = ReachG\X (r) for some r ∈ V (G),
• V1 (A) consists of triples (X, X 0 , R) ∈ V1 (A) for all (X, R) ∈ V0 (A) and all
X 0 ∈ Lc ,
• For all (X, R) ∈ V0 (A) and all X 0 ∈ Lc there is an edge in E(A) from (X, R)
to (X, X 0 , R), and
• If R0 = ReachG\X 0 (r0 ) then there is an edge in E(A) from (X, X 0 , R) to
(X 0 , R0 ) if, and only if, r0 ∈ ReachG\(X∩X 0 ) (R).
Remark. In the sequel, it may be more convenient to view (non-initial) positions of the
game as pairs (X, r) with X ⊆ V (G) and r ∈ V (G) to represent the position (X, R)
where R = ReachG\X (r).
We recall from Chapter 5 the definitions of a search and a strategy. As with the
game characterizing tree-width, we are interested in the minimum number of cops
required to capture the robber. Because of this, and from the definition of the game, it
follows that we may assume the first move of the cops is to not place any cops on the
graph and “wait and see” where the robber moves: if the robber can win from (∅, r1 )
for some r1 ∈ V (G) then he can win from (∅, V (G)), and conversely, if the cops have
a winning strategy σ which uses at most k cops from (∅, r) for all r ∈ V (G), then the
strategy defined by playing ∅ at (∅, V (G)) and σ otherwise is also a winning strategy
which uses at most k cops. In view of this, and the above remark, we introduce a more
practical definition of a strategy where the strategy is only defined for positions (X, r)
where X ⊆ V (G), |X| ≤ k, and r ∈ V (G).
Definition 6.2 (k-cop strategy). Let G be a directed graph, and consider the cops and
visible robber game on G. A (k-cop) strategy for the cops is a function σ : [V (G)]≤k ×
V (G) → [V (G)]≤k . A search (X1 , r1 )(X2 , r2 ) · · · is consistent with a strategy σ if
Xi+1 = σ(Xi , ri ) for all i. A strategy σ is a winning strategy, if every search consistent
with σ is finite.
In a similar way, we can define a strategy for the robber against k cops.
Definition 6.3 (Strategy against k cops). Let G be a directed graph, and consider
the cops and visible robber game on G. A strategy against k cops is a function
ρ : [V (G)]≤k × [V (G)]≤k × V (G) → V (G) such that for all X, X 0 ⊆ V (G) and
r ∈ V (G) \ X, ρ(X, X 0 , r) ∈ ReachG\(X∩X 0 ) (r). A search (X1 , r1 )(X2 , r2 ) · · · is
consistent with a strategy ρ if ri+1 = ρ(Xi , Xi+1 , ri ) for all i.
We observe that, similar to the game on undirected graphs, variants of the cops and
visible robber game where only one cop can be moved at a time, or the cops are lifted
and placed in separate moves are all equivalent in that the number of cops required to
capture the robber on a graph does not depend on the variant.
We call the graph searching width (recall Definition 5.36) associated with this game
and the resource we are interested in bounding, the cop number of the graph. That is,
Definition 6.4 (Cop number). The cop number of a directed graph G is the least k
such that k cops have a strategy to win the cops and visible robber game on G.
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CHAPTER 6. DAG-WIDTH
Before we introduce the technical aspects of this game needed in later sections, we
present a couple of results that illustrate some of its properties.
Lemma 6.5. Let G be a (finite) non-empty directed graph. At least one cop is required
to capture a visible robber on G and exactly one cop is required if, and only if, G is
acyclic.
Proof. As we have no requirement that the robber moves, as long as there is one vertex,
the robber can defeat zero cops by remaining at that vertex. That is, if v ∈ V (G), then
function ρ defined by ρ(∅, ∅, v) = v is clearly a winning strategy against 0 cops.
If G is acyclic, then one cop can catch the robber by always playing to the current position of the robber. Eventually, the robber will not be able to move and the
cops will capture him. More precisely, define σ(X, r) = {r}. Then for any search
(X0 , r0 )(X1 , r1 ) · · · consistent with σ, we observe that for all i, ri 6= ri+1 and there
is a directed path from ri to ri+1 . Since G is finite and acyclic, it follows that every
search consistent with σ must be finite and therefore winning for the cops.
Conversely, if G has a cycle (v1 , v2 , . . . , vm ), then the robber can defeat one cop
by forever staying in the cycle. That is, for all r ∈ V (G) and X ∈ [V (G)]≤1 let
ρ(X, X 0 , r) = v1 for all X 0 such that v1 ∈
/ X 0 and ρ(X, {v1 }, r) = v2 . This is clearly
a strategy for the robber against one cop, and as any search consistent with ρ can be
extended to an infinite search, it is winning for the robber.
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The cops and visible robber games we have already seen Chapter 5 characterizing
tree-width and directed tree-width have the property that they are invariant under edge
reversal. That is, the number of cops required to catch the robber does not change if
the directions of all the edges of the graph are reversed. As we see below, this is not
the case for the game we consider here. One exception is graphs of cop number 1, that
is, acyclic graphs. We recall from Section 1.1.2, the definition of G op .
Proposition 6.6. The cop number of a directed graph G is 1 if, and only if, the cop
number of G op is 1.
Proof. This follows from Lemma 6.5 by observing that G is acyclic if, and only if, G op
is acyclic.
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Proposition 6.7. For any j, k with 2 ≤ j ≤ k, there exists a graph Tkj with cop
number j such that the cop number of (Tkj )op is k.
Proof. Informally, Tkj is a binary branching tree of height k such that every vertex
has edges to all its descendants, and edges back to its j − 1 nearest ancestors.1 More
precisely, Tkj is the directed graph defined as follows:
• V (Tkj ) = {w ∈ {0, 1}∗ : |w| < k}, and
• (w1 , w2 ) ∈ E(Tkj ) if, and only if, either w1 ≺ w2 or w2 ≺ w1 and |w1 |−|w2 | <
j, where ≺ is the prefix ordering on {0, 1}∗ .
1 To aid informal descriptions we view this graph as a directed tree with additional structure. Thus we
use descendants, ancestors, root and leaves to refer to various vertices in the graph as they would be in the
underlying directed tree.
6.1. COPS AND VISIBLE ROBBER GAME
107
We now show that the cop number of Tkj is j and the cop number of (Tkj )op is k.
First we see that j cops have a winning strategy on Tkj by initially playing on the
root then following the robber down, in a leap-frogging manner, whichever subtree he
plays in. More precisely, we inductively define the strategy σ as follows. Initially,
σ(∅, V (G)) = {}. We observe that from the definition of the edge relation, if the
robber chooses to respond by moving to a vertex w with first symbol 0, then he is
unable to reach any vertex w0 with first symbol 1. Similarly if the robber chooses to
move to a vertex with first symbol 1, he cannot reach any vertex in the 0-subtree. Now
suppose the cops are on X and the robber is on wr and X and wr satisfy the following:
There exists wmin and wmax such that X = {w : wmin w ≺ wmax } and
ReachT j \X (wr ) = {w : wmax w}.
(∗)
k
Then wmax is the next vertex to be occupied by a cop. If |X| < j, then σ(X, wr ) =
X ∪ {wmax }, otherwise if |X| = j, σ(X, wr ) = X \ {wmin } ∪ {wmax}. Let wr0 be the
next location of the robber after the cops move from X to X 0 = σ(X, wr ). We show
that the resulting position (X 0 , wr0 ) satisfies (∗). Clearly from the definition of σ, we
have either X 0 = {w : wmin w wmax } or X 0 = {w : wmin ≺ w wmax }, so the
first part of (∗) is true. Next we show that wr0 ∈ ReachT j \X (X)wr \ {wmax }. Clearly,
k
if X 0 ⊇ X this is true, so we need only consider the case when |X| = j. But this
implies |wmax | − |wmin | = j, thus there are no edges from wmax to wmin . As wmin
is the only vertex vacated and every vertex reachable from wr is reachable from wmax ,
the set of vertices reachable by the robber must decrease. Now let w0 be the shortest
word which is a prefix of wr0 and for which wmax is a proper prefix. It follows from the
definition of the edge relation that every vertex which the robber can reach must have
w0 as a prefix. Thus ReachT j \X (X 0 )wr0 = {w : w0 w}. Clearly the strategy σ is a
k
strategy for j cops, we now show that it is winning. We observe that for every search
consistent with σ, the sequence of wmax is a sequence of words of increasing length.
So after k moves there will be no vertex available for the robber to move to. Thus σ is a
winning strategy for j cops. A winning strategy for k cops on (Tkj )op can be similarly
defined, replacing j with k in the above definition. Note that when |X| = k there is no
vertex available for the robber, so the cops never have to make a “leap-frog” move.
We now show that the robber can defeat j − 1 cops on Tkj and k − 1 cops on (Tkj )op .
The strategy for the robber involves choosing some leaf. Whenever a cop moves to that
leaf, a simple counting argument shows that there must be at least one unoccupied
ancestor which the robber can reach with at least one clear path to a leaf below. The
robber then plays to that ancestor and along that path to the leaf. More precisely, let
L = {w ∈ V (Tkj ) : |w| = k − 1}. For each X, X 0 ∈ [V (G)]<j and wr ∈ V (G),
let ρ(X, X 0 , wr ) = w0 for some w0 ∈ (L ∩ ReachT j \X ((X ∩ X 0 ))r) \ X 0 . Clearly
k
if ρ is well defined, it describes a winning strategy for the robber against j − 1 cops.
We now show that there always exists some such w0 . Since |L| = 2k−1 > j − 1, the
robber can always choose an element of L initially, so we may assume that wr ∈ L. If
wr ∈
/ X 0 then choosing w0 = wr suffices, so suppose wr ∈ X 0 . Since wr ∈
/ X and
|X|, |X 0 | < j, it follows that |X ∩ X 0 | < j − 1. Thus there exists w00 ≺ wr such that
|wr | − |w00 | < j and {w : w00 w and wr0 6 w} ∩ X 0 = ∅ where wr0 is the shortest
108
CHAPTER 6. DAG-WIDTH
word which is a prefix of wr and for which w00 is a proper prefix. Thus for every w ∈ L
such that w00 is a prefix of w, there is a path from wr to w in Tkj \ (X ∩ X 0 ). Thus
choosing w0 ∈ L such that w00 is a prefix of w gives a well-defined strategy. A winning
strategy for the robber against k − 1 cops on (Tkj )op is defined similarly, replacing j
with k in the above definition.
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6.1.1 Monotonicity
For the remainder of this chapter, we are primarily concerned with monotone strategies.
We recall from Definition 5.8 the definitions of fugitive-monotone (robber-monotone)
and searcher-monotone (cop-monotone) searches and strategies. We observe that, as
with the cops and visible robber game on undirected graphs, the cops and visible
robber game for directed graphs permits idling and is vacating sensitive. Thus from
Lemma 5.11, we have:
Lemma 6.8. A cop-monotone winning strategy for k cops is robber-monotone.
We saw in Theorem 5.37 that for the cops and visible robber game on undirected
graphs, the converse to this holds: if k cops have a robber-monotone winning strategy
then k cops have a cop-monotone winning strategy. In [JRST01] it was shown that this
is not the case for the strongly connected visible robber game. The next result shows
that as with the game on undirected graphs, for the game we are considering, the two
notions of monotonicity coincide.
Lemma 6.9. If k cops have a cop-monotone or robber-monotone winning strategy,
then they have a winning strategy that is both cop-monotone and robber-monotone.
Proof. From Lemma 6.8, it suffices to show that if k cops have a robber-monotone
winning strategy then k cops have a cop-monotone winning strategy. Suppose the cops
have a robber-monotone winning strategy, and let (X0 , r0 )(X1 , r1 ) · · · be a search
consistent with that strategy. From this we construct a sequence which can be used
to define a cop-monotone strategy in the obvious way. Suppose Xi 6⊆ Xi+1 and let
v ∈ Xi \ Xi+1 . As v ∈ Xi , the robber is unable to reach v when the cops are on
Xi . As the strategy is robber-monotone, the robber is unable to reach v at any further
stage, in particular, he cannot reach v when the cops are on Xi+1 . Thus, no cop needs
to revisit v in order to prevent the robber from reaching v. Thus, we can remove v
from all Xj , j > i. Proceeding in this way results in a sequence (X0 , r0 )(X10 , r1 ) · · · .
0
The strategy which takes (Xi0 , ri ) to Xi+1
is cop-monotone for this search. Repeating
this for all plays (that is, every choice for robber) results in a cop-monotone strategy.
Hence, whenever the cops have a robber-monotone winning strategy they also has a
cop-monotone strategy.
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With this lemma in mind we define a monotone winning strategy in the obvious way.
Note that we have actually proved a slightly stronger assertion:
Corollary 6.10. If k cops have a monotone winning strategy in the cops and visible robber game on a digraph G, then k cops have a winning strategy σ such that
σ(X, r) ⊆ X ∪ ReachG\X (r) for all X ⊆ V (G) and r ∈ V (G) \ X.
6.2. DAG-DECOMPOSITIONS AND DAG-WIDTH
109
In Theorem 5.37, we also saw that in the visible robber game on undirected graphs,
if k cops have a winning strategy then k cops have a monotone winning strategy. An
interesting question is whether this extends to the game on directed graphs. Kreutzer
and Ordyniak [KO07] have recently shown that this is not the case.
Theorem 6.11 ([KO07]). For any m ∈ N, there exists a digraph for which 5m cops can
capture a visible robber but 6m cops are required to do so with a monotone strategy.
Of course, this result does not preclude the possibility that, as with the strong visible
robber game, the number of cops required for a monotone capture is bounded by some
function of the number of cops required for a winning strategy which is not necessarily
monotone. This gives us the following interesting open problem:
Open problem 6.12. Does there exist a function f : ω → ω such that for all digraphs
G, if k cops can capture a visible robber on G then f (k) cops can capture the robber
with a monotone strategy?
6.2 DAG-decompositions and DAG-width
In this section, we present a decomposition of directed graphs that is somewhat similar
in style to tree decompositions of undirected graphs. This leads to the definition of
DAG-width, which can be seen as a measure of how close a given graph is to being
acyclic. We show then that a graph has DAG-width k if, and only if, k cops have a
monotone winning strategy in the cops and robber game played on that graph. We
conclude with some algorithmic properties enjoyed by DAG-width.
Definition 6.13 (Guarding). Let G be a directed graph. A set W ⊆ V (G) guards a
set V ⊆ V (G) if W ∩ V = ∅ and whenever there is an edge (u, v) ∈ E(G) such that
u ∈ V and v 6∈ V , then v ∈ W .
Definition 6.14 (DAG-decomposition). Let G be a digraph. A DAG-decomposition
of G is a pair (D, X ) where D is a directed, acyclic graph and X = (Xd )d∈V (D) is a
family of subsets of V (G) such that
S
(D1) d∈V (D) Xd = V (G).
(D2) For all vertices d D d0 D d00 , Xd ∩ Xd00 ⊆ Xd0 .
(D3) S
For all edges (d, d0 ) ∈ E(D), Xd ∩ Xd0 guards X≥d0 \ Xd , where X≥d0 :=
d0 D d00 Xd00 . For any root d, X≥d is guarded by ∅.
The width of a DAG-decomposition (D, X ) is defined as max{|Xd | : d ∈ V (D)}. The
DAG-width of a graph is defined as the minimal width of any of its DAG-decompositions.
The main result of this section is an equivalence between monotone strategies for
the cop player and DAG-decompositions.
Theorem 6.15. For any directed graph G, there is a DAG-decomposition of G of
width k if, and only if, k cops have a monotone winning strategy in the cops and visible
robber game on G.
CHAPTER 6. DAG-WIDTH
110
To prove this, we first need some simple observations about guarding.
Lemma 6.16. Let G be a directed graph, and W, X, Y, Z ⊆ V (G).
(i) X guards ReachG\X (Y ).
(ii) If W guards Y , X guards Z, then (W ∪ X) \ (Y ∪ Z) guards Y ∪ Z.
(iii) If X guards Y , Z ⊇ X and Z ∩ Y = ∅, then Z guards Y .
(iv) If X guards Y then X ∪ Z guards Y \ Z
Proof. (i): Clearly X ∩ ReachG\X (Y ) = ∅. Now suppose (v, w) ∈ E(G), v ∈
ReachG\X (Y ) and w ∈
/ ReachG\X (Y ). It follows from the definition of ReachG\X (Y )
that w ∈ X. Therefore X guards ReachG\X (Y ).
(ii): Suppose (v, w) ∈ E(G), v ∈ Y ∪ Z and w ∈
/ Y ∪ Z. If v ∈ Y , then
w ∈ W , as W guards Y . Similarly, if v ∈ Z then w ∈ X as X guards Z. Hence
w ∈ (W ∪ X) \ (Y ∪ Z), and (W ∪ X) \ (Y ∪ Z) guards Y ∪ Z.
(iii): Suppose (v, w) ∈ E(G), v ∈ Y and w ∈
/ Y . As X guards Y , w ∈ X. As
Z ⊇ X, w ∈ Z. Therefore, Z guards Y .
(iv): Since X ∩ Y = ∅ and Z ∩ (Y \ Z) = ∅, it follows that (X ∪ Z) ∩ (Y \ Z) = ∅.
Now suppose (v, w) ∈ E(G), v ∈ Y \ Z and w ∈
/ Y \ Z. Thus, w ∈
/ Y or w ∈ Z. For
the first case, w ∈ X as X guards Y . Hence w ∈ X ∪ Z.
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We now turn to the proof of Theorem 6.15.
Proof of Theorem 6.15. Suppose k cops have a monotone winning strategy σ in the
cops and visible robber game on a directed graph G. As σ is monotone, from Corollary 6.10 it follows that we may assume that cops are only ever placed on vertices that
are reachable by the robber. That is,
σ(X, r) ⊆ X ∪ ReachG\X (r).
(6.1)
We recall the definition of a strategy DAG, Dσ , from Definition 5.7. Since the
nodes of Dσ are positions in the cops and robber game, the function σ is well defined
for all d ∈ V (Dσ ). We claim that (Dσ , X ), with X defined by Xd = σ(d) for all
d ∈ V (Dσ ), is a DAG-decomposition of G of width ≤ k. To support our claim, we
first observe the following simple facts. For d = (X, r) ∈ V (Dσ ),
[
ReachG\X (r) ⊆
σ(d0 ) ⊆ X ∪ ReachG\X (r).
(6.2)
dDσ d0
The first inclusion follows from the fact that σ is a winning strategy for the cop
player: at position (X, r) every vertex reachable by the robber (ReachG\X (r)) will
be occupied by a cop at some point in the future. The second inclusion follows from
repeated application of (6.1). Further, for d = (X, r) ∈ V (Dσ ),
ReachG\X (r) = ReachG\(X∩σ(X,r)) (r).
(6.3)
6.2. DAG-DECOMPOSITIONS AND DAG-WIDTH
111
As X ∩ σ(X, r) ⊆ X, ReachG\X (r) ⊆ ReachG\(X∩σ(X,r)) (r). The reverse inclusion
follows from the fact that σ is a robber-monotone strategy.
Equations (6.2) and (6.3) together imply for d = (X, r):
[
σ(d0 ) \ X = ReachG\(X∩σ(X,r)) (r).
(6.4)
dDσ d0
We now show that (Dσ ,SX ) is indeed a DAG-decomposition of width ≤ k. For (D1),
if there was a v ∈ V (G) \ d∈V (Dσ ) Xd , then the robber could defeat σ by playing to
S
v at the beginning and staying there indefinitely. Hence d∈V (D) Xd = V (G). (D2)
follows immediately from the (cop-)monotonicity of the winning strategy σ. Towards
establishing (D3), let us first consider a root d = (X, r) of Dσ . From the definition of
Dσ , this root is unique, thus X≥d = V (G) and is therefore guarded by ∅. Now suppose
(d, d0 ) ∈ E(Dσ ). If d0 = (X 0 , r0 ) then Xd = σ(d) = X 0 . So by (6.4),
[
X≥d0 \ Xd =
σ(d00 ) \ X 0 = ReachG\(X 0 ∩σ(X 0 ,r0 )) (r0 ).
d0 Dσ d00
Therefore, from Lemma 6.16(i), Xd ∩ Xd0 = X 0 ∩ σ(X 0 , r0 ) guards X≥d0 \ Xd . It
follows that (Dσ , X ) is a DAG-decomposition. To see that it has width ≤ k, note that
max{|Xd | : d ∈ V (Dσ )} = max{|σ(d)| : d ∈ V (Dσ )} ≤ k.
Conversely, let (D, X ) be a DAG-decomposition of width k. A strategy for k cops
can then be defined as:
(1) Let the robber choose a vertex v ∈ V (G). From (D1), there exists dv ∈ V (D) such
that v ∈ Xdv . Let d be a root of D which lies above dv .
(2) Place cops on Xd .
(3) From (D3) and Lemma 6.16(iii), Xd guards X≥d \ Xd . Therefore, the robber can
only move to vertices in X≥d \ Xd . Suppose the robber moves to v 0 ∈ Xd00 . Let
d0 be a successor of d which lies above d00 .
(4) Remove cops on Xd \ Xd0 (leaving cops on Xd ∩ Xd0 )
(5) As Xd ∩ Xd0 guards X≥d0 \ Xd , the robber can only move to vertices in X≥d0 –
that is, the robber must remain in the sub-DAG rooted at d0 .
(6) Return to step 2 with d0 as d.
As D is a DAG, at some point the robber will not be able to move because X≥d \ Xd
is empty when d is a leaf. Hence, this is a winning strategy for k cops. To show that it
is monotone, observe that (D2) ensures that at no point does a cop return to a vacated
vertex. This concludes the proof of Theorem 6.15.
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We observe that as a strategy DAG is the underlying DAG in the decomposition
(D, X ) constructed in this proof, and a strategy DAG has a unique root, we have the
following:
112
CHAPTER 6. DAG-WIDTH
Corollary 6.17. If a digraph G has a DAG-decomposition of width k, then G has a
DAG-decomposition (D, X ) of width ≤ k such that D has a unique root.
In the sequel we show that we can make further simplifying assumptions about the
structure of DAG-decompositions.
The remainder of this section looks at some properties of DAG-decompositions
motivated by similar results for tree-width and tree decompositions. We first observe
that the winning strategies for the cop player in Lemma 6.5 and Proposition 6.7 are
monotone. These results therefore imply that a graph has DAG-width 1 if, and only if,
it is acyclic (indeed, the graph itself will suffice as a decomposition) and that the DAGwidth of a graph may change by an arbitrary amount if its edges are reversed. This
last observation is particularly useful when searching for alternative characterizations
of DAG-width, such as those we introduce in Chapter 8.
We further observe that, as with the game on undirected graphs, the cops and visible
robber game enjoys the properties of graph searching games introduced in Section 5.4.
In particular this means that DAG-width decreases when taking subgraphs, and suitably
increases when taking lexicographic products.
Lemma 6.18. Let (D, X ) be a DAG-decomposition of a digraph G, and let G 0 be a subgraph of G. (D, X |G 0 ) where X |G 0 := Xd ∩ V (G 0 ) d∈V (D) is a DAG-decomposition
of G 0 .
Proof. Clearly, (D1) and (D2) still hold for (D, X |G 0 ). For (D3), we observe that,
if X guards Y in G, then X ∩ V (G 0 ) guards Y ∩ V (G 0 ) in G 0 . This is because, if
v ∈ Y ∩ V (G 0 ), w ∈ V (G 0 ) \ Y and (v, w) ∈ E(G 0 ) ⊆ E(G), then w ∈ X (as X
guards Y ), hence w ∈ X ∩ V (G 0 ). Then, (D3) follows immediately from (D3) for the
original decomposition (D, X ).
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t
Corollary 6.19. Let G and G 0 be directed graphs such that G 0 is a subgraph of G. Then
DAG-width(G 0 ) ≤ DAG-width(G).
Lemma 6.20. Let G be a directed graph and Kn the complete graph on n vertices.
DAG-width(G • Kn ) = n · DAG-width(G).
Proof. From Theorem 5.64, it suffices to show that the cops and visible robber game is
composition-expanding. We show that it satisfies conditions (I)–(IV) of Lemma 5.65.
Clearly as the cops are free to make any move, conditions (I) and (II) are satisfied. For
condition (III), suppose on G as the cops move from X to X 0 , the robber can move
from r to r0 . It follows by the definitions of Reach and lexicographic product that if the
cops move from X ×V (Kn ) to X 0 ×V (Kn ) in G •Kn , the robber can move from (r, v)
to (r0 , w0 ) for all v, w ∈ V (Kn ). Thus there is an edge in the arena (of the game on
G • Kn ) from (X × V (Kn ), X 0 × V (Kn ), R × V (Kn )) to (X 0 × V (Kn ), R0 × V (Kn ))
where R = ReachG\X (r) and R0 = ReachG\X 0 (r0 ). Finally, to show condition (IV),
we observe that for X ⊆ V (G • Kn ) and (r, v) ∈ V (G • Kn ), Reach(G•Kn )\X (r, v)
consists of those vertices (r0 , v 0 ) ∈
/ X such that r0 inReachG\Y (r) where Y = {x ∈
V (G) : (x, w) ∈ X for all v ∈ V (Kn )}. Thus, if there is an edge in the arena (for the
game on G) from (Y, Y 0 , S) to (Y 0 , S 0 ), then there is an edge in the arena (for the game
on G • Kn ) from (X, X 0 , R) to (X 0 , R0 ) where X, X 0 , Y , Y 0 , R, R0 , S and S 0 are as
defined in condition (IV) of Lemma 5.65.
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6.2. DAG-DECOMPOSITIONS AND DAG-WIDTH
113
We also show that the DAG-width of graphs is closed under directed unions, which,
as we discussed in Chapter 4, is an important property of a reasonable decomposition
of directed graphs.
Lemma 6.21. Let G be a directed union of the digraphs G1 and G2 . Then
DAG-width(G) = max{DAG-width(G1 ), DAG-width(G2 )}.
Proof. For DAG-decompositions (D1 , X 1 ) and (D2 , X 2 ) of G1 and G2 respectively,
the DAG D obtained by adding an edge from every leaf of D1 to every root of D2 .
˙ d2 )d∈V (D2 ) forms a DAG-decomposition of G.
together with X := (Xd1 )d∈V (D1 ) ∪(X
Conversely, any DAG-decomposition (D, X ) of G can be restricted to G1 and G2 yielding DAG-decompositions for these graphs, according to Lemma 6.18.
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t
We observe that it follows that the DAG-width of a directed graph is the maximum
DAG-width of all its strongly connected components.
For algorithmic purposes, it is often useful to have a normal form for decompositions. The following is similar to one for tree decompositions as presented in [Bod97].
Definition 6.22. [Nice DAG-decompositions] A DAG-decomposition (D, X ) is nice
if
(N1) D has a unique root.
(N2) Every d ∈ V (D) has at most two successors.
(N3) If d1 , d2 are two successors of d0 , then Xd0 = Xd1 = Xd2 .
(N4) If d1 is the unique successor of d0 , then |Xd0 4 Xd1 | ≤ 1.
The final result we establish in this section is that every graph with DAG-width k
has a nice decomposition with width k. For this, we transform a DAG-decomposition
into one which is nice that has the same width. To do this we formalize the transformations we use, and show that executing them (possibly subject to some constraints)
does not violate any of the properties of a DAG-decomposition. First we require the
following useful observation.
Lemma 6.23. Let (D, X ) be a DAG-decomposition. For all (d, d0 ) ∈ E(D),
X≥d0 \ Xd = X≥d0 \ (Xd ∩ Xd0 ).
Proof. As Xd ∩ Xd0 ⊆ Xd , X≥d0 \ Xd ⊆ X≥d0 \ (Xd ∩ Xd0 ). Conversely, suppose
v ∈ X≥d0 , that is, v ∈ Xd00 for some d00 D d0 . We will show that v ∈ Xd ∩ Xd0 , or
v∈
/ Xd . Suppose v ∈ Xd . Then as d D d0 D d00 , v ∈ Xd ∩ Xd00 ⊆ Xd0 . Hence
v ∈ Xd ∩ Xd0 . Thus, X≥d0 \ Xd ⊇ X≥d0 \ (Xd ∩ Xd0 ).
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t
Definition 6.24 (Splitting). Let (D, X ) be a DAG-decomposition, and suppose d0 ∈
V (D) has m > 1 successors d1 , d2 , . . . , dm . The decomposition (D0 , X 0 ) obtained
from (D, X ) by splitting d0 is defined as follows:
˙ l , dr },
(i) V (D0 ) = V (D)∪{d
CHAPTER 6. DAG-WIDTH
114
Xd0
Xd1
Xd2
Xd0
44
44
44
44
44
44
...
Xdm
Xdl
44
44
44
Xdr
Xd1
Xd2
...
44
44
44
Xdm
Figure 6.1: Splitting at d0
(ii)
E(D0 ) =
E(D) \ {(d0 , di ) : 1 ≤ i ≤ m}
∪ {(d0 , dl ), (d0 , dr ), (dl , d1 )}
∪ {(dr , di ) : 2 ≤ i ≤ m}, and
(iii) Xd0 = Xd , for all d ∈ V (D), and Xd0 l = Xd0 r = Xd0 .
Figure 6.1 gives a visual representation of this transformation.
Lemma 6.25. Let (D, X ) be a DAG-decomposition of a digraph G of width k, and
suppose d0 ∈ V (D) has m > 1 successors d1 , d2 , . . . , dm . Then (D0 , X 0 ) obtained
from (D, X ) by splitting d0 is a DAG-decomposition of G of width k.
Proof. First we observe that, as d0 is the unique predecessor of dl and dr , for any
d ∈ V (D) such that d ≺D0 dl or d ≺D0 dr , it must be the case that d D d0 . Thus, for
all d ∈ V (D),
[
[
0
=
X≥d
Xd0 0 =
Xd0 = X≥d ,
dD0 d0
dD d0
since if Xdl or Xdr is included in the union on the left, then so is Xd0 , and so neither
Xdl nor Xdr contribute to the overall union.
Also, for all i such that 1 ≤ i ≤ m, it is the case that Xd0 ∩Xdi guards X≥di \ Xd0 .
Therefore, by Lemma 6.16(iii),
Xd0 guards X≥di \ Xd0 .
(6.5)
0
ItSis easily seen that
Sthe edges added do not create any cycles, so D is a DAG. Fur0
ther, d∈V (D0 ) Xd = d∈V (D) Xd = V (G). To prove the connectivity condition (D2),
let d, d0 , d00 ∈ V (D0 ), be such that d D0 d0 D0 d00 . If d0 = d or d00 then trivially
Xd0 ∩ Xd0 00 ⊆ Xd0 0 , so suppose d ≺D0 d0 ≺D0 d00 . We consider four cases:
• If none of d, d0 , d00 is dl or dr , then d, d0 , d00 ∈ D, and (D2) follows from the fact
that (D, X ) is a DAG-decomposition.
• If d is dl or dr then since all descendants of d are in V (D), and d0 ∈ V (D)
is the unique predecessor of d, we obtain the following chain of nodes in D:
d0 ≺D d0 ≺D d00 . So Xd0 ∩ Xd0 00 = Xd0 ∩ Xd00 ⊆ Xd0 = Xd0 0 .
6.2. DAG-DECOMPOSITIONS AND DAG-WIDTH
115
• If d00 is dl or dr then from the comments at the beginning of the proof, d ≺D
d0 D d0 . Thus, Xd0 ∩ Xd0 00 = Xd ∩ Xd0 ⊆ Xd0 = Xd0 0 .
• Finally, if d0 is dl or dr then by the same reasoning as the previous two cases,
d D d0 ≺D d00 . So Xd0 ∩ Xd0 00 = Xd ∩ Xd00 ⊆ Xd0 = Xd0 0 .
Thus, in all cases, Xd0 ∩ Xd0 00 ⊆ Xd0 0 , showing that (D2) holds. To see that condition
(D3) also holds, observe first that every root of D0 is a root of D too. So ∅ guards
0
X≥d = X≥d
. Now let (d, d0 ) ∈ E(D0 ). We consider three cases:
• d0 ∈ V (D) (i.e., d0 6= dl , dr ). If d = dl or dr , then Xd0 = Xd0 . Otherwise
0
0
(d, d0 ) ∈ E(D). In both cases, Xd0 ∩ Xd0 0 guards X≥d
0 \ Xd .
0
0
0
• d0 = dl (so d = d0 ). Here X≥d
0 = Xd0 ∪ X≥d1 , so X≥d0 \ Xd = X≥d1 \ Xd0 .
0
0
0
Hence, by (6.5), Xd0 = Xd ∩ Xd0 guards X≥d1 \ Xd0 = X≥d0 \ Xd0 .
S
0
0
• d0 = drS(so d = d0 ). HereSX≥d
0 = Xd 0 ∪
2≤i≤m X≥di , and so X≥d0 \
0
Xd = ( X≥di ) \ Xd0 = (X≥di \ Xd0 ), where the unions are taken over
iSfor 2 ≤ i ≤ m. From Lemma 6.16(ii) and (6.5), Xd0 ∩ Xd0 0 = Xd0 guards
0
0
2≤i≤m (X≥di \ Xd0 ) = X≥d0 \ Xd .
As Xd0 l = Xd0 r = Xd0 , we have
max{|Xd0 | : d ∈ V (D0 )} = max{|Xd | : d ∈ V (D)} = k.
Consequently, the decomposition (D0 , X 0 ) has width k.
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t
By the decomposition resulting from splitting d m − 1 times we mean the decomposition resulting from splitting d, and then recursively splitting dr until dr has only
one successor. A complete split of (D, X ) is the decomposition (D0 , X 0 ) obtained by
recursively splitting every node with more than two successors.
Definition 6.26 (Adding). Let (D, X ) be a DAG-decomposition of a digraph G. If
(d0 , d1 ) ∈ E(D) and X ⊆ V (G) the decomposition resulting from adding X to
(d0 , d1 ) is the pair (D0 , X 0 ) with
˙ X}
(i) V (D0 ) = V (D)∪{d
(ii) E(D0 ) = (E(D) \ {(d0 , d1 )}) ∪ {(d0 , dX ), (dX , d1 )}
(iii) Xd0 X = X, and for all d ∈ V (D), Xd0 = Xd .
See Figure 6.2 for a visual interpretation.
Lemma 6.27. Let (D, X ) be a DAG-decomposition of a digraph G of width k and let
(D0 , X 0 ) be the decomposition resulting from adding X ⊆ V (G) to (d0 , d1 ). If either
(i) Xd0 ∩ Xd1 ⊆ X ⊆ Xd0 , or
(ii) Xd0 ∩ Xd1 ⊆ X ⊆ Xd1 ,
then (D0 , X 0 ) is a DAG-decomposition of G of width k.
CHAPTER 6. DAG-WIDTH
116
Xd0
Xd0
+
X
X
Xd1
Xd1
Figure 6.2: Adding X to (d0 , d1 )
Proof. We observe that for all d ∈ V (D), if d ≺D0 dX , then, as d0 ∈ V (D) is the
unique predecessor of dX , we have d D d0 , and if dX ≺D0 d, then as d1 ∈ V (D) is
the unique successor of dX , we have d1 D d. This implies, for all d ∈ V (D)
[
[
0
X≥d
=
Xd0 0 =
Xd0 = X≥d ,
dD0 d0
dD d0
since if Xd0 X is included in the union on the left, then both Xd0 0 and Xd0 1 are, and so in
either case of the lemma Xd0 X = X does not contribute to the overall union.
Further, Xd0 ∩ Xd1 guards X≥d1 \ Xd0 = X≥d1 \ (Xd0 ∩ Xd1 ) from Lemma 6.23.
Clearly, D0 is a DAG. We now
(D0 , X 0 )Ssatisfies the properties (D1)
S show that
0
to (D3). It is easily seen that d∈V (D0 ) Xd = X ∪ d∈V (D) Xd = V (G). This
shows (D1). Towards establishing condition (D2), suppose d D0 d0 D0 d00 . If
d0 = d or d0 = d00 then trivially Xd0 ∩ Xd0 00 ⊆ Xd0 0 , so suppose d ≺D0 d0 ≺D0 d00 . We
consider four cases:
• If none of d, d0 , d00 is dX then d, d0 , and d00 are all in V (D), so (D2) follows from
the fact that (D, X ) is a DAG-decomposition.
• Suppose d = dX . From the observations made at the beginning of the proof,
we get the following chain of nodes in D: d0 ≺D d1 D d0 ≺D d00 . So in
case (i) of the lemma, we have X ⊆ Xd0 . So Xd0 ∩ Xd0 00 = X ∩ Xd00 ⊆
Xd0 ∩Xd00 ⊆ Xd0 = Xd0 0 , by condition (D2) of (D, X ). Otherwise, if X ⊆ Xd1 ,
then Xd0 ∩ Xd0 00 = X ∩ Xd00 ⊆ Xd1 ∩ Xd00 ⊆ Xd0 = Xd0 0 .
• The other cases are similar. If d00 = dX then we obtain d ≺D d0 D d0 ≺D d1 .
So if X ⊆ Xd0 , then Xd0 ∩ Xd0 00 = Xd ∩ X ⊆ Xd ∩ Xd0 ⊆ Xd0 = Xd0 0 . If
X ⊆ Xd1 , then Xd0 ∩ Xd0 00 = Xd ∩ X ⊆ Xd ∩ Xd1 ⊆ Xd0 = Xd0 0 .
• Finally, assume d0 = dX . Then d D d0 ≺D d1 D d00 . Hence Xd ∩ Xd00 ⊆
Xd0 and Xd ∩ Xd00 ⊆ Xd1 . Thus, Xd0 ∩ Xd0 00 = Xd ∩ Xd00 ⊆ Xd0 ∩ Xd1 ⊆ X =
Xd0 0 .
Finally, towards (D3), if d is a root of D0 , then d is a root of D. Hence ∅ guards
X≥d = X≥d0 . Now let (d, d0 ) ∈ E(D0 ). We consider three cases:
• dX 6∈ {d, d0 }, i.e., (d, d0 ) ∈ E(D). In this case, (D3) follows from the fact that
(D, X ) is a DAG-decomposition.
6.2. DAG-DECOMPOSITIONS AND DAG-WIDTH
117
• Now suppose d = dX (so d0 = d1 ). If Xd0 ∩ Xd1 ⊆ X ⊆ Xd0 , so we are in
case (i) of the lemma, then
X≥d1 \ (Xd0 ∩ Xd1 )
⊇
X≥d1 \ X
⊇
X≥d1 \ Xd0 .
Further, by Lemma 6.23, X≥d1 \ (Xd0 ∩ Xd1 ) = X≥d1 \ Xd0 . Therefore X≥d1 \
X = X≥d1 \ Xd0 . As (D, X ) is a DAG-decomposition, Xd0 ∩ Xd1 guards
X≥d1 \ Xd0 , and as Xd0 ∩ Xd1 ⊆ X ∩ Xd1 , Lemma 6.16(iii) implies that
0
Xd0 ∩ Xd0 1 = X ∩ Xd1 guards X≥d1 \ Xd0 = X≥d
\ Xd0 .
1
Otherwise we are in case (ii) and we have Xd0 ∩ Xd1 ⊆ X ⊆ Xd1 . Let Z =
X \ (Xd0 ∩ Xd1 ). We know (Xd0 ∩ Xd1 ) guards X≥d1 \ (Xd0 ∩ Xd1 ), due to
Lemma 6.23. Hence, by Lemma 6.16(iv), Xd0 ∩ Xd0 1 = X = (Xd0 ∩ Xd1 ) ∪ Z
guards
(X≥d1 \ (Xd0 ∩ Xd1 )) \ Z = X≥d1 \ ((Xd0 ∩ Xd1 ) ∪ Z)
0
\ Xd0 0 .
= X≥d1 \ X = X≥d
1
0
• Finally, suppose d0 = dX (so d = d0 ). Here we claim X≥d
\ Xd0 0 = X≥d1 \
X
0
0
Xd0 . If X ⊆ Xd0 , then X≥dX \ Xd0 = (X ∪ X≥d1 ) \ Xd0 = (X \ Xd0 ) ∪
0
=
(X≥d1 \ Xd0 ) = X≥d1 \ Xd0 . If X ⊆ Xd1 , then since dX D0 d1 , X≥d
X
0
0
X≥d1 = X≥d1 . Now X ⊇ Xd0 ∩ Xd1 , so by Lemma 6.16(iii), Xd0 = X guards
0
\ Xd0 0 .
X≥d1 \ Xd0 = X≥d
X
Note that since X ⊆ Xd0 or Xd1 , max{|Xd0 | : d ∈ V (D0 )} = max{|Xd | : d ∈
V (D)} = k. So (D0 , (Xd0 )d∈V (D0 ) ) has width k.
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t
If X1 , X2 , . . . , Xn is a sequence of subsets of V (G), the decomposition resulting
from adding X1 , X2 , . . . , Xn to (d0 , d1 ) is the decomposition resulting from adding
X1 to (d0 , d1 ) and then recursively adding Xi+1 to (dXi , d1 ).
We can now describe how to transform a DAG-decomposition into one which is
nice and has the same width.
Theorem 6.28. If G has a DAG-decomposition of width k, then G has a nice DAGdecomposition of width k.
Proof. Let (D, X ) be a DAG-decomposition of width k. From Corollary 6.17, we may
assume that D has a unique root. We carry out each of the following steps.
1. We apply a complete split on (D, X ) to obtain a DAG-decomposition such that
every node has at most two successors, and if d has two successors d1 and d2 ,
then Xd = Xd1 = Xd2 . This establishes (N2) and (N3).
2. To satisfy (N4), we require two stages. First, for each (d0 , d1 ) ∈ E(D) with
Xd0 6= Xd1 , we add Xd0 ∩ Xd1 to (d0 , d1 ) to obtain a DAG-decomposition such
that for every (d, d0 ) ∈ E(D0 ), Xd is either a subset or a superset of Xd0 .
3. Secondly, for each (d, d0 ) ∈ E(D) with |Xd |−|Xd0 | = m > 1 (or |Xd0 |−|Xd | =
m > 1), let X0 = Xd , X1 , . . . , Xm = Xd0 be a strictly decreasing (increasing)
118
CHAPTER 6. DAG-WIDTH
sequence of subsets. Such a sequence exists because at the previous step we
finished with a DAG-decomposition such that Xd ⊆ Xd0 or Xd ⊇ Xd0 . Add
X1 , X2 , . . . , Xm−1 to (d, d0 ). At this point we have a decomposition which
satisfies (N1) to (N4), and is therefore nice.
Finally, from Lemmas 6.25 and 6.27, at each step we have a DAG-decomposition of
width k.
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6.3 Algorithmic aspects of DAG-width
We now consider algorithmic applications of DAG-width as well as the complexity of
deciding the DAG-width of a graph and computing a DAG-decomposition.
6.3.1 Computing DAG-width and decompositions
Because deciding if the tree-width of a graph is at most a given integer is NP-complete,
it is no surprise that deciding if the DAG-width of a graph is at most a given integer is
intractable. Indeed, the following is a direct consequence of the NP-completeness of
the T REE - WIDTH decision problem and Proposition 6.36.
Theorem 6.29. Given a digraph G and a natural number k, deciding if the DAG-width
of G is at most k is NP-hard.
Despite the similarity to tree-width, it is currently unknown whether deciding if the
DAG-width of a graph is bounded by a given value is in NP. However, we strongly
believe that this is the case, giving us the following:
Conjecture 6.30. Given a digraph G and a natural number k, deciding if the DAGwidth of G is at most k is NP-complete.
However, for any fixed k, it is possible, in polynomial time, to decide if a graph
has DAG-width at most k and to compute a DAG-decomposition of this width if it has.
This follows in a similar manner to Proposition 5.71, so for the proof of the next result
we refer the reader to Section 5.5.
Theorem 6.31. Let G be a directed graph and let k < ω. Deciding if k cops have a
monotone winning strategy in the cops and visible robber game on G, and computing
such a strategy if it exists can be executed in time O(|V (G)|2k+4 ).
Note also that the translation of strategies into decompositions is computationally
easy, that is, it can be done in polynomial time. Since winning strategies can be computed in polynomial time in the size of the graph, we get the following.
Proposition 6.32. Given a graph G of DAG-width k, a DAG-decomposition of G of
width k can be computed in time O(|G|O(k) ).
6.3. ALGORITHMIC ASPECTS OF DAG-WIDTH
119
6.3.2 Algorithms on graphs of bounded DAG-width
We can use DAG-decompositions, particularly nice DAG-decompositions, to define dynamic programming algorithms similar to those used with tree decompositions. Working bottom-up from the leaves of the underlying DAG D, for each node d ∈ V (D)
we
S compute a data set containing information for the subgraph induced by X≥d :=
d0 D d Xd . The general pattern is described in Algorithm 6.1. We observe that if the
starting decomposition is nice, then the combine and expand steps become significantly
simplified. Indeed, the combine step can be seen as applying to inner nodes with two
successors and the update steps apply to inner nodes with only one successor.
Algorithm 6.1 Dynamic programming using a DAG-decomposition
Given a DAG-decomposition (D, X ):
Leaves: Compute the data set for Xd for all leaves d.
Combine: If d ∈ V (D) is an inner node with successors d1 , . . . , dm , S
combine the data
m
sets computed for X≥d1 , . . . , X≥dm to a data set for the union i=1 X≥di .
Expand: Finally, expand the data set to include Xd .
As the directed tree-width of a graph is bounded above by a constant factor of its
DAG-width (see Proposition 6.37), any graph property that can be decided in polynomial time on classes of graphs of bounded directed tree-width can be decided on
classes of graphs of bounded DAG-width also. This implies that properties such as
Hamiltonicity that are known to be polynomial time on graphs of bounded directed
tree-width can be solved efficiently on graphs of bounded DAG-width too. We give a
nontrivial application of DAG-width in Section 6.3.3 where we show that parity games
can be solved efficiently on arena of bounded DAG-width, something which is not
known for directed tree-width.
We observe that the arena used in the proof of Theorem 2.64 has DAG-width 2:
place one cop on vertex qϕ and the remaining graph is acyclic and can be searched
monotonely with one cop. This implies that, unlike parity games, win-set games (and,
consequently, Muller games, Zielonka DAG games, Emerson-Lei games and circuit
games) remain hard on arenas of bounded DAG-width.
Proposition 6.33. Deciding win-set games on arenas of DAG-width 2 is P SPACE -hard.
As for the relation to undirected tree-width, it is clear that not all graph properties
that can be decided in polynomial time on graphs of bounded tree-width can also be
decided efficiently on graphs of bounded DAG-width. For instance, the 3-colourability
problem is known to be decidable in polynomial time on graphs of bounded tree-width.
However, the problem does not depend on the direction of edges. For any given (undirected) graph, we can simply direct the edges in such a way that it becomes acyclic.
Thus, arbitrary instances are polynomial-time reducible to instances of DAG-width 1.
As 3-colourability over arbitrary graphs is NP-hard, it follows that the problem cannot
be solved in polynomial time on graphs of bounded DAG-width, unless P TIME = NP.
120
CHAPTER 6. DAG-WIDTH
6.3.3 Parity Games on Graphs of Bounded DAG-Width
Using the algorithm scheme of Algorithm 6.1, we now outline a dynamic programming
algorithm for solving parity games. The advantage of such an algorithm is that on any
class of arenas of bounded DAG-width it runs in polynomial time, giving us a large
class of graphs for which there exists a tractable algorithm for solving parity games.
Full details of the algorithm can be found in [BDHK06].
Given an arena A, a DAG-decomposition of A is a DAG-decomposition of the
underlying directed graph (V (A), E(A)).
Theorem 6.34. For any k, given a parity game (A, χ) where the DAG-width of A is at
most k, determining if Player 0 has a winning strategy can be decided in polynomial
time.
Let us fix a parity game (A, χ) where χ : V (A) → P, and let n = |V (A)|. We
assume that every vertex in A has out-degree at most 2. It is easy to see that the arena
resulting from the transformation described in Theorem 2.59, replacing vertices that
have out-degree more than 2 with binary branching trees, requires at most one more
cop to capture a visible robber. Thus such a transformation results in an arena with
DAG-width at most k + 1. Let (D, X ) be a DAG-decomposition of A of width k which
we assume is nice. For technical reasons, we also assume that for the root d of D,
Xd = ∅. From Proposition 6.32 we can compute such a decomposition in polynomial
time. The idea is that we utilise the restrictions imposed by a DAG-decomposition
to bound the number of strategies we need to consider. Although memoryless strategies are sufficient for parity games, we do not assume the strategies we consider are
memoryless.
Consider U ⊆ V (A) and a set W that guards U . Fix a pair of strategies σ and τ .
For any v ∈ U , there is exactly one play π = v0 v1 · · · that is consistent with Player 0
playing σ and Player 1 playing τ . Let π 0 be the maximal prefix of π that is contained
in U . The outcome of the pair of strategies (σ, τ ) (given U and v) is defined as follows.


if π 0 = π and π is winning for Even;
win0
outσ,τ (U, v) := win1
if π 0 = π and π is winning for Odd;


(vi+1 , p) if π 0 = v0 · · · vi and p = max{χ(vj ) : j ≤ i + 1}.
That is to say that, if the play consistent with Player 0 playing σ and Player 1 playing
τ leads to a cycle contained entirely within U , then the outcome simply records which
player wins the game. However, if the winner is not determined entirely within U , the
outcome records the vertex w in W in which the play emerges from U and the largest
priority that is seen in the play π starting in v and ending in w, including the end points.
By construction, if outσ,τ (U, v) = (w, p) then w ∈ W . More generally, for any set
W ⊆ V , define the set of potential outcomes in W , written pot-out(W ), to be the set
{win0 , win1 } ∪ {(w, p) : w ∈ W and p ∈ P}.
We recall from Chapter 3, the definition of the reward order v. We now define a
partial order E on pot-out(W ) which orders potential outcomes according to how good
they are for Player 1. It is the least partial order satisfying the following conditions:
(i) win1 E o for all outcomes o;
6.3. ALGORITHMIC ASPECTS OF DAG-WIDTH
121
(ii) o E win0 for all outcomes o;
(iii) (w, p) E (w, p0 ) if p v p0 for all w ∈ W .
In particular, (w, p) and (w0 , p0 ) are incomparable if w 6= w0 . The idea is that if τ and
τ 0 are strategies such that outσ,τ (U, v)Eoutσ,τ 0 (U, v) then Player 1 is better off playing
strategy τ rather than τ 0 in response to Player 0 playing according to σ.
A single outcome is the result of fixing the strategies played by both players in
the subgame induced by a set of vertices U . If we fix the strategy of Player 0 to
be σ but consider all possible strategies that Player 1 may play, we can order these
strategies according to their outcome. If one strategy achieves outcome o and another o0 with o E o0 , there is no reason for Player 1 to consider the latter strategy.
Thus, we define resultσ (U, v) to be the set of outcomes that are achieved by the best
strategies that Player 1 may follow, in response to Player 0 playing according to σ.
More formally, resultσ (U, v) is the set of E-minimal elements in the set {o : o =
outσ,τ (U, v) for some τ }. Thus, resultσ (U, v) is an anti-chain in the partial order (pot-out(W ), E),
where W is a set of guards for U . Finally, we write R ESULT(U, v) for the set {resultσ (U, v) :
σ is a strategy for Player 0}.
The data structure which we wish to compute is defined as follows. For any d ∈
V (D), let Vd = X≥d \ Xd . Let
F RONTIER (d) = {(v, r) : v ∈ Vd and r ∈ R ESULT(Vd , v)}.
We show how to compute in polynomial time F RONTIER (d) for all d ∈ V (D). It
follows from the definitions that if win0 ∈ R ESULT(V (A), v), then Player 0 has a
winning strategy from v. Thus, as X≥r = V (A) when r is the root of D, it follows
that win0 ∈ R ESULT(X≥r , vI (A)) if, and only if, Player 0 wins the game.
We observe that Since X≥d \ Xd is guarded by Xd , |Xd | ≤ k and |Vd | ≤ n, the
number of distinct values of resultσ (Vd , v) as σ ranges over all possible strategies is
at most (n + 1)k + 2. This bound on the number of possible values of resultσ (Vd , v)
guarantees that |F RONTIER (d)| ≤ n (n + 1)k + 2 .
We now outline how we compute F RONTIER (d) for each stage of the dynamic
programming scheme presented earlier.
Leaves: If d ∈ V (D) is a leaf, then as |Vd | ≤ k, it is clear that for all v ∈ Vd ,
R ESULT(Vd , v), and hence F RONTIER (d), can be computed in constant time.
Combine: If d ∈ V (D) is a node with two successors d1 and d2 , then as Xd = Xd1 =
Xd2 , it follows that Vd = Vd1 ∪ Vd2 . In this case, as Xd guards Vd1 and Vd2
there is no path from a vertex in Vd1 to a vertex in Vd2 except through Xd . It is
straightforward to show that F RONTIER (d) = F RONTIER (d1 ) ∪ F RONTIER (d2 ).
Expand: If d ∈ V (D) is a node with one successor d0 , we consider three cases.
Case 1: Xd = Xd0 . In this case, F RONTIER (d) = F RONTIER (d0 ).
Case 2: Xd \ Xd0 = {u}. Then, by (D2), u 6∈ Vd0 . Also, by the definition of
Vd , u 6∈ Vd . We conclude that Vd = Vd0 . Moreover, since Xd0 guards Vd0 (by
Lemma 6.16(iii)), there is no path from any element of Vd0 to u except through
122
CHAPTER 6. DAG-WIDTH
Xd0 . Thus, if (w, p) ∈ resultσ (Vd , v) for some v and σ, it must be the case that
w ∈ Xd0 . Hence, F RONTIER (d) = F RONTIER (d0 ).
Case 3: Xd0 \ Xd = {u}. This is the critical case. Here Vd = Vd0 ∪ {u} and
in order to construct F RONTIER (d) we must determine the results of all plays
beginning at u. If u has one successor, then this is trivial, so let us assume u
has 2 successors u1 and u2 . We observe that for i ∈ {1, 2} either ui ∈ Xd
or ui ∈ Vd0 . If ui ∈ Xd , let Ri = {(ui , max{p, q})}, where p = χ(u) and
q = χ(ui ). Otherwise let Ri = R ESULT(Vd0 , ui ). Thus Ri is the set of outcomes
obtained if the play proceeds from u to ui .
Consider a play from v ∈ Vd . If it does not reach u, then we can read, from
R ESULT(Vd0 , v) ∈ F RONTIER (d0 ), the outcome of the play. Otherwise, if the
play reaches u, it continues to either u1 or u2 . If both u1 and u2 are in Vd0 then
either the play returns to u, in which case we know the winner of the play, or the
play reaches a vertex in Xd . This latter case also occurs if either of u1 or u2 is
in Xd . Thus to compute R ESULT(Vd , v), and hence F RONTIER (d), we proceed
as follows.
For each r ∈ R ESULT(Vd0 , v), we do the following. If there is no p ∈ P such
that (u, p) ∈ r add r to R ESULT(Vd , v). Otherwise, let (u, p) ∈ r for some
p. We now consider two cases. If u ∈ V1 (A) then for each r1 ∈ R1 and
r2 ∈ R2 , let R = r1 ∪ r2 . Replace each (w, q) ∈ R with (w, max{p, q}). Let
R0 = R ∪ (r \ {(u, p)}). If (u, q) ∈ R0 for some odd q then Player 1 wins
the play for the chosen strategies, so replace (u, q) with win1 . Similarly, replace
(u, q) ∈ R0 for q even with win0 . Finally, we remove the elements of R0 which
are not E-minimal and add R0 to R ESULT(Vd , v).
Now suppose u ∈ V0 (A) for each r0 ∈ R1 ∪ R2 , if (u, q) ∈ r0 and max p, q
0
is odd, replace
let R =
r with win1 and0 add it0 to R ESULT(Vd , v). 0 Otherwise,
r \ {(u, p)} ∪ (w, q) : (w, q ) ∈ r and q = max{p, q } . If R contains a
pair (u, q) then q must be even and we replace this pair in R by win0 . Finally,
we add the E-minimal elements of R to R ESULT(Vd , v).
In a similar way, we can also compute the set R ESULT(Vd , u).
It is clear from the bounds on the size of F RONTIER (d) that at each stage, F RONTIER (d)
can be computed in polynomial time. Since the DAG-decomposition has size at most
O(n2k+4 ), it follows that this algorithm runs in polynomial time. This completes the
outline of the proof of Theorem 6.34.
6.4 Relation to other graph connectivity measures
As a structural measure for undirected graphs, the concept of tree-width is of unrivalled robustness. On the realm of directed graphs, however, its heritage seems to be
split among several different concepts. In the sequel we compare DAG-width with
several other connectivity measures for directed graphs, namely tree-width, directed
tree-width, and directed path-width. We show that, despite their similar nature, the
measures are all significantly different.
6.4. RELATION TO OTHER GRAPH CONNECTIVITY MEASURES
123
6.4.1 Undirected tree-width
First we formalize the relationship between DAG-width and undirected tree-width alluded to in previous sections. We recall from Chapter 4, the definition of tree-width.
We also recall that the tree-width of a directed graph G is defined as the tree-width of
the undirected graph obtained from G by forgetting the orientation of the edges.
Proposition 6.35.
(i) If a directed graph G has tree-width k, it has DAG-width at most k + 1.
(ii) There exists a family of directed graphs with arbitrarily large tree-width and
DAG-width 1.
Proof. (i): Suppose (T , W) is a tree decomposition of G of width k, with W =
(Wt )t∈V (T ) . Choose some r ∈ V (T ) and orient the edges of T away from r. That is, if
{s, t} ∈ E(T ) and s is on the unique path from r to t, then change {s, t} to (s, t). Since
T is a tree, every edge has a unique orientation in this manner. Let D be the resulting
DAG. For all d ∈ V (D), set Xd := Wt where t is the node of T corresponding to d.
We claim that (D, X ) with X = (Xd )d∈V (D) is a DAG-decomposition of G of width
k + 1. The condition (D1) is trivial from (T1); (D2) follows from (T2). The orientation
ensures D has one root r, so X≥r = V (G). Condition (D3) is hence satisfied at the
root. For the other nodes, (D3) follows from Lemma 4.2. Let (d, d0 ) ∈ E(D) and
/ X≥d0 \ Xd .
suppose v ∈ X≥d0 \ Xd . Suppose also that (v, w) ∈ E(G) and w ∈
As there is a path (of length 1) from v to w, it follows from Lemma 4.2 that either
/ Xd , w ∈ Xd ∩ Xd0 and (D3) holds.
v ∈ Xd ∩ Xd0 or w ∈ Xd ∩ Xd0 . Since v ∈
(ii): For any integer n ∈ N, let Kn be the (undirected) complete graph with n
vertices v1 , v2 , . . . , vn . Orient the edges of Kn such that (vi , vj ) is an edge if and only
if i < j. The resulting directed graph is acyclic and therefore has DAG-width 1, but
the underlying undirected graph is a complete graph of n vertices and therefore has
tree-width n − 1.
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We now observe that DAG-width is equivalent to tree-width on undirected graphs
if we view an undirected graph as a directed graph in the natural way. We recall from
←
→
Section 1.1.2, the directed graph G obtained from an undirected graph G by replacing
each edge {u, v} with two anti-parallel edges (u, v) and (v, u).
Proposition 6.36. Let G be an undirected graph. G has tree-width k − 1 if, and only
←
→
if, G has DAG-width k.
Proof. It is easily seen that the k-cops and robber game for undirected graphs on G is
←
→
equivalent to the k-cops and robber game for directed graphs on G . The result follows
from the correspondence between the measures and existence of monotone winning
strategies.
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6.4.2 Directed tree-width
In Chapter 4 we saw directed tree-width from [JRST01] and in Chapter 5 we discussed
how it was characterized by the strong visible robber game. We can use this game characterization to relate directed tree-width and DAG-width: as the strong visible robber
124
CHAPTER 6. DAG-WIDTH
game is defined similarly to the cops and visible robber game with added restrictions
on movement of the robber, we see that a (robber-monotone) winning strategy for k
cops in the cops and visible robber game is a (robber-monotone) winning strategy for
k cops in the strong visible robber game. Thus, we can use Lemma 5.41 to obtain a
bound on the directed tree-width. Towards a converse to this, we show that directed
tree-width and DAG-width are very different measures by exhibiting a class of graphs
with small directed tree-width and arbitrarily large DAG-width.
Proposition 6.37.
(i) If a directed graph G has DAG-width k, it has directed tree-width at most 3k + 1.
(ii) There exists a family of graphs with arbitrarily large DAG-width and directed
tree-width 1.
Proof. (i): If G has DAG-width k then k cops can win the cops and visible robber game
on G. Thus, k cops can win the strongly visible robber game on G, as the robber is more
restricted in this game. From Lemma 5.41, it follows that G has directed tree-width at
most 3k + 1.
(ii): Consider the family {(Tk2 )op : k ≥ 2} of graphs defined in Proposition 6.7.
Note that (Tk2 )op is a binary branching tree of height k with back-edges from every
vertex to each of its ancestors. We have shown that (Tk2 )op has cop number k, and
it is clear that the strategy described for k cops is monotone, so (Tk2 )op has DAGwidth k. On the other hand, consider the directed tree T obtained from (Tk2 )op by
removing back-edges. For each t0 ∈ V (T ), let Bt0 := {t, s} where t is the vertex
in V ((Tk2 )op ) corresponding to t0 and s is the predecessor of t (if t0 is not the root
of T ), and let W(s0 ,t0 ) := {s} for all (s0 , t0 ) ∈ E(T ). Then, it is easy to see that
(T , (Bt0 )t0 ∈V (T ) , (We )e∈E(T ) ) is a directed tree decomposition of (Tk2 )op of width 1.
For k ≥ 2, (Tk2 )op is not acyclic and therefore has directed tree-width exactly 1.
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6.4.3 Directed path-width
We saw in Chapter 4 the definition of path-width. According to Barát [Bar05], Reed,
Seymour and Thomas defined a natural extension of path-width to directed around
1995, however [Tho02] seems to be the first occurrence of the definition in the literature. The definition mirrors the definition of path-width, however the direction of the
edges is accounted for by fully utilising the linear ordering present in a sequence.
Definition 6.38 (Directed path decompositions and directed path-width [Bar05]). Let
G be a directed graph. A directed path decomposition of G is a sequence X1 , . . . , Xn
of subsets of V (G) such that:
Sn
(DP1) i=1 Xi = V (G),
(DP2) If i ≤ j ≤ k then Xi ∩ Xk ⊆ Xj , and
(DP3) For each e = (u, v) ∈ E(G), there exists i ≤ j such that u ∈ Xi and v ∈ Xj .
6.4. RELATION TO OTHER GRAPH CONNECTIVITY MEASURES
125
The width of a directed path decomposition, X1 , . . . , Xn , is max{|Xi | : 1 ≤ i ≤
k} − 1. The directed path-width of G is the smallest width of any directed path decomposition of G.
Just as tree-width can be characterized by the cops and visible robber game, we
saw in Chapter 5 that path-width can also be characterized by a cops and robber game:
the cops and invisible robber game of Example 5.2.2. In [Bar05] Barát considered the
natural extension of this cops and robber game to directed graphs and showed that it the
number of cops required to capture an invisible robber lies within one of the directed
path-width of the graph. He also observed that the number of cops required to capture
an invisible robber with a cop-monotone strategy is equal to one more than the directed
path-width of the graph.
It is therefore not surprising that directed path-width is intimately related to DAGwidth. From the game characterizations, it appears that directed path-width is to DAGwidth as path-width is to tree-width. Indeed, as we see from the definitions the two
are closely connected. In fact, a DAG-decomposition can be seen as a generalization
of a directed path decomposition where we replace the linear order of the subsets of
V (G) with a partial order. This means that a directed path decomposition is a DAGdecomposition where the underlying DAG is a directed path. It is therefore not surprising that DAG-width bounds directed path-width below and there are families of graphs
of bounded DAG-width and unbounded directed path-width. Just as the class of binary
trees is a class of graphs with bounded tree-width and unbounded directed path-width,
we now show that the class of bidirected binary trees is a class of graphs with bounded
DAG-width and unbounded directed path-width.
Proposition 6.39.
(i) If a directed graph G has directed path-width k, it has DAG-width at most k + 1.
(ii) There exists a family of graphs with arbitrarily large directed path-width and
DAG-width 2.
Proof. (i): Let W1 , . . . , Wn be a directed path decomposition of G of width k. Let
Dn be the directed path with n vertices. That is V (Dn ) = {d1 , . . . , dn } and (di , dj ) ∈
E(Dn ) if, and only if, j = i + 1. Set Xdi := Wi for all di ∈ V (Dn ). We claim
(Dn , (Xd )d∈V (Dn ) ) is a DAG-decomposition of G of width k + 1. Condition (D1)
follows from (DP1) and (D2) follows from (DP2). To show (D3) for 1 ≤ i < n,
suppose v ∈ X≥di+1 \ Xdi and (v, w) ∈ E(G). From (DP3) there exist i0 ≤ j 0 such
that v ∈ Wi0 and w ∈ Wj 0 . If i0 ≤ i, then by (DP2) v ∈ Xdi , contradicting the choice
/ X≥di+1 \ Xdi then w ∈ Xdi and
of v. Thus, i < i0 ≤ j 0 and w ∈ X≥di+1 . If w ∈
therefore w ∈ Xdi+1 by (DP2). Thus, Xdi ∩ Xdi+1 guards X≥di+1 \ Xdi .
(ii): Let Tk be the (undirected) binary tree of height k ≥ 2. From Proposition 6.36,
←
→
←
→
Tk has DAG-width 2. It is easy to see that on Tk , an invisible robber can defeat
←
→
k − 1 cops, but k cops have a winning strategy. Therefore, from [Bar05], Tk must
←
→
have directed path-width at least k − 2. Thus, the family { Tk : k ≥ 2} satisfies the
proposition.
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126
CHAPTER 6. DAG-WIDTH
Chapter 7
Digraph measures: Kelly-width
In Chapter 4 we introduced the concept of tree-width as a measure of graph complexity.
We remarked on its usefulness for algorithmic purposes, and discussed the importance
of the problem of extending tree-width to directed graphs. In this chapter, we continue
investigating this extension by considering other characterizations of tree-width and
their natural generalizations to digraphs.
Part of the reason why tree-width is such a good measure of graph complexity is
that many other measures arising from different areas of graph theory can be shown
to be equivalent to tree-width. For instance, we saw in Chapter 5 that the number of
cops required to capture a visible robber in a graph-searching game is equivalent to the
tree-width of that graph. In this chapter we consider three other characterizations of
tree-width: partial k-trees, elimination orders and a graph searching game in which an
invisible robber attempts to avoid capture by a number of cops, subject to the restriction that he may only move if a cop is about to occupy his position. Partial k-trees are
the historical forerunner of tree-width and are therefore associated with graph structure
theory [Ros70]. In fact, many of the original algorithmic results for tree-width were
formulated in terms of partial k-trees (see, for example [AP89]). Elimination orderings are particularly useful in the analysis of (symmetric) matrix factorizations such
as Cholesky decompositions [Liu90]. For example, elimination orders can be used
to determine the parallel time required to factorize a symmetric matrix with Gaussian
elimination [BGHK95]. Finally, as we saw in Chapter 5 (and also [DKT97, FHT04]),
graph searching games have recently been used to explore and generate robust measures of graph complexity. We generalize all these to directed graphs, resulting in partial k-DAGs, directed elimination orderings, and an inert robber game on digraphs. We
show that all these generalizations are equivalent on digraphs and are also equivalent
to the width-measure associated to a new kind of decomposition we introduce. As the
game is reminiscent of capturing hideout-based outlaws, we propose the name Kellydecompositions, after the infamous Australian bushranger Ned Kelly. The fact that all
these notions are equivalent on digraphs as they are on undirected graphs suggests that
this might be a robust measure of complexity and connectivity of digraphs.
As with tree-decompositions and DAG-decompositions, Kelly-decompositions have
a structure that is well suited for designing dynamic programming algorithms that will
127
128
CHAPTER 7. KELLY-WIDTH
run in polynomial time when the width of these decompositions is bounded. However, unlike DAG-decompositions (as far as is currently known), the size of Kellydecompositions can be made linear in the size of the graph it decomposes, significantly
reducing the space complexity of such algorithms. As with the previous chapter, we
will introduce a general scheme for producing dynamic programming algorithms that
use the additional structural information provided by Kelly-decompositions. We illustrate its use by producing algorithms for solving NP-complete problems such as
Hamiltonian cycle, and computing the winner of a parity game. Both these algorithms
run in polynomial time on graphs of bounded Kelly-width.
The chapter is organised as follows. In the first section we formally define inert
robber games, elimination orders, and partial k-trees and k-DAGs. We show that on
digraphs the associated width measures are all equivalent. In Section 7.2, we introduce Kelly-decompositions and Kelly-width and show that it also coincides with the
measures defined in Section 7.1. In Section 7.3, we present applications: Algorithms
for Hamiltonian cycle, weighted disjoint paths and parity games that all run in polynomial time on graphs of bounded Kelly-width, and detail a connection between Kellydecompositions and asymmetric matrix factorization. Finally, we compare Kelly-width
to some of the other directed graph measures we have already seen such directed treewidth and DAG-width, showing that it is a unique measure of complexity. However,
we also provide evidence to suggest that Kelly-width and DAG-width are measuring
the same fundamental property of digraphs.
7.1 Inert robber games, elimination orderings, and partial k-DAGs
7.1.1 Inert robber game
The cops and robber game we consider for this chapter is the cops and inert robber
game from Example 5.2.5. This game consists of an invisible robber who is able to run
from his position along any path which does not pass through a cop, however he may
only move if a cop is about to land on his position. For convenience, we say that he is
inert. The natural generalization of this game to directed graphs is defined as followed.
Definition 7.1 (Cops and inert robber game). Let G be a directed graph. The cops and
inert robber game on G is the cops and robber game defined by (Lc , Lr , A), where
• Lc = P(V (G)) and Lr = P(V (G)) \ {∅},
• V0 (A) consists of pairs (X, R) ∈ Lc × Lr such that X ∩ R = ∅,
• V1 (A) consists of triples (X, X 0 , R) ∈ V1 (A) for all (X, R) ∈ V0 (A) and all
X 0 ∈ Lc ,
• For all (X, R) ∈ V0 (A) and all X 0 ∈ Lc there is an edge in E(A) from (X, R)
to (X, X 0 , R), and
7.1. GAMES, ORDERINGS AND K -DAGS
129
• There is an edge in E(A) from (X, X 0 , R) to (X 0 , R0 ) if, and only if,
R0 = R ∪ ReachG\(X∩X 0 ) (X 0 ∩ R) \ X 0 .
We recall from Chapter 5 the definitions of a search, monotonicity and a strategy.
As with the game characterizing tree-width, we are interested in the minimum number
of cops required to capture the robber, so we also recall the definition of a strategy
for k cops from Definition 5.32. Since R0 is uniquely defined from X, R and X 0 ,
the inert robber game is in actuality a single player game. As we mentioned earlier,
this is typical for games with an invisible robber. One consequence is that given a
strategy for the cops, there is a unique play consistent with that strategy. We call this
the play associated with the strategy. In the remainder of this chapter we are primarily
concerned with robber-monotone strategies. However, we first show that the added
constraint on the movement of the invisible robber does not affect the existence of a
cop-monotone winning strategy for k cops.
Proposition 7.2. Let G be a digraph. Then k cops have a cop-monotone winning
strategy in the cops and invisible robber game on G if, and only if, k cops have a
cop-monotone winning strategy in the cops and inert robber game on G.
Proof. Since the cops and inert robber game is more restrictive on the robber than the
cops and invisible robber game, a winning strategy in the latter is a winning strategy
in the former. We now show how a cop-monotone winning strategy, σ, for k cops in
the cops and inert robber game is also a cop-monotone winning strategy for k cops in
the cops and invisible robber game. Let (X0 , R0 )(X1 , R1 ) · · · (Xn , Rn ) be the unique
search associated with σ in the cops and inert robber game. We define a k-cop copmonotone strategy, σ 0 , for the cops and invisible robber game as follows. Define Ri0
0
inductively as: R00 = V (G), and for 1 ≤ i ≤ n, Ri0 = ReachG\(Xi ∩Xi−1 ) (Ri−1
) \ Xi .
0
0
0
Then define σ (Xi , Ri ) = Xi+1 , so σ is essentially the strategy resulting from playing
σ in the cops and invisible robber game. By definition, (X0 , R00 )(X1 , R10 ) · · · is the
search associated with σ 0 , and it is clearly a cop-monotone strategy for k cops. We
now show that it is winning. In particular, we prove by induction on i that Ri0 = Ri for
0 ≤ i ≤ n.
Since R0 = V (G) = R00 our claim is clearly true for i = 0. Now suppose Ri = Ri0
for some i ≥ 0. Since Ri ∪ ReachG\(Xi ∩Xi+1 ) (Ri ∩ Xi+1 ) ⊆ ReachG\(Xi ∩Xi+1 ) (Ri0 ),
0
0
we have Ri+1 ⊆ Ri+1
. So suppose Ri+1 6⊇ Ri+1
. Then there exists w ∈ ReachG\(Xi ∩Xi+1 ) (Ri )\
Xi+1 such that w ∈
/ Ri ∪ ReachG\(Xi ∩Xi+1 ) (Ri ∩ Xi+1 ) \ Xi+1 . Thus w ∈
/ Xi ∪ Ri .
Note that since w ∈
/ Ri , we have i ≥ 1. Furthermore, there exists v ∈ Ri \ Xi+1
such that there is a path from v to w in G \ (Xi ∩ Xi+1 ). Let v 0 be the last element of Ri on this path, and let w0 ∈
/ Ri be the successor of v 0 on this path. Since
0
the path is in G \ (Xi ∩ Xi+1 ), w ∈
/ Xi ∩ Xi+1 . Suppose w0 ∈
/ Xi . Then since
Xi ∩ Ri = ∅, there is a path from v to w0 in G \ Xi . Therefore, as v ∈ Ri = Ri0 =
ReachG\(Xi−1 ∩Xi ) (Ri−1 ) \ Xi we have w0 ∈ Ri , contradicting the definition of w0 .
Thus
w0 ∈ Xi \ Xi+1 .
Now let j ≥ i + 1 be such that v 0 ∈ Rj \ Rj+1 . Since σ is winning, and v 0 ∈ Ri+1 ,
CHAPTER 7. KELLY-WIDTH
130
there is such a j. By the definition of the cops and inert robber game, it must be that
v 0 ∈ Xj+1 \Xj . We claim that w0 ∈ Rj+1 . Since i ≤ i+1 ≤ j+1, and w ∈ Xi \Xi+1 ,
by the cop-monotonicity of σ, w0 ∈
/ Xj+1 . Therefore, as (v 0 , w0 ) ∈ E(G),
w0 ∈ ReachG\(Xj ∩Xj+1 ) (Rj ∩ Xj+1 ) = Rj+1 .
Now let l ≥ j + 1 be such that w0 ∈ Rl \ Rl+1 . Since σ is winning and w0 ∈ Rj+1 ,
such an l exists. By the definition of the cops and inert robber game, it must be that
w0 ∈ Xl+1 . But since i ≤ i + 1 ≤ l + 1, and w0 ∈ Xi , by the cop-monotonicity of σ,
w0 ∈ Xi+1 – contradiction. Thus Ri ⊇ Ri0 , and therefore Ri = Ri0 .
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7.1.2 Elimination orderings
Our next definition extends the idea of vertex elimination to digraphs. Vertex elimination, for undirected and directed graphs, has been researched for many years in the
study of linear programming [RT75]. One technique for solving a system of equations
is to combine equations so that the value of some variables can easily be determined,
thereby eliminating those variables and reducing the system to a simpler one. This
elimination process may introduce new relations between the remaining variables, and
capturing this process in a more general setting is the motivation behind vertex elimination of graphs. We can represent a system of equations as a graph with a vertex for
each variable occurring in the system, and an edge between variables that are related
by some equation in the system. Vertex elimination is then a symbolic representation
of variable elimination.
More precisely, let G be an undirected graph and v ∈ V (G). To eliminate v from
G, we remove v, but add edges (if necessary) between any two vertices adjacent to v.
In this way, we see that vertex elimination is the process of removing vertices from a
graph but adding edges to preserve reachability. It is this concept that we extend to
directed graphs.
Definition 7.3 (Directed elimination). Let G be a digraph and v ∈ V (G). The graph
resulting from directed elimination of v from G is the graph G 0 obtained from G by
deleting v and adding new edges (u, w) (if necessary) if (u, v) and (v, w) ∈ E(G).
We can use vertex elimination to define a complexity measure on undirected graphs.
Let G be an undirected graph. A linear order C = (v1 , v2 , . . . , vn ) on V (G) defines
a sequence of eliminations whereby the vertices of G are successively eliminated in
the order specified by C. For convenience we call C an elimination ordering and this
sequence of eliminations, the elimination defined by C. We define the width of C to be
the maximum of the degrees of the vertices when they are eliminated. These definitions
easily translate to directed graphs, but the complexity measure we are interested in is
the maximum out-degree of eliminated vertices.
Definition 7.4 ((Partial) Directed elimination ordering). Let G be a digraph and let
V ⊆ V (G) be a subset of vertices. A partial directed elimination ordering on V is a
linear ordering C = (v1 , v2 , . . . , vn ) of V . A directed elimination ordering is a partial
directed elimination ordering on V (G). The (partial) directed elimination defined by
7.1. GAMES, ORDERINGS AND K -DAGS
131
C
be
C is the following sequence of directed graphs. We define G0C := G, and let Gi+1
C
the graph resulting from directed elimination of vi+1 from Gi . The width of C is the
maximum over all i of the out-degree of vi in GiC . For convenience we also define the
support of vi with respect to C as suppC (vi ) := {vj : (vi , vj ) ∈ E(GiC )}.
We observe that the width of a directed elimination ordering is the maximum cardinality of all its supports.
Immediately from the definitions, we have this simple lemma relating the support of
an element in an elimination ordering to the set of vertices reachable from that vertex.
Lemma 7.5. Let C be a directed elimination ordering of a graph G and let v ∈ V (G).
Let R := {u : v C u}. Then suppC (v) = {u : v C u and there is v 0 ∈ ReachG\R (v)
such that (v 0 , u) ∈ E(G)}.
7.1.3 Partial k-trees and partial k-DAGs
The class of k-trees and, more generally, chordal graphs are important and widely
studied classes of undirected graphs. A graph is chordal if any cycle of four or more
vertices contains a chord – an edge between a pair of vertices not adjacent in the cycle,
and a chordal graph is a k-tree if it contains no (k + 2)-clique as a subgraph. These
structural restrictions are algorithmically beneficial: for example, chordal graphs have
a linear number of maximal cliques, so problems such as finding a clique of a given
size, which are in general NP-complete, can be efficiently solved on chordal graphs
and k-trees.
An equivalent way to characterize the class of k-trees is as a class of graphs generated by a generalization of how one might construct a tree.
Definition 7.6 ((Partial) k-trees). The class of k-trees is defined recursively as follows:
• The complete graph on k vertices is a k-tree.
• A k-tree with n+1 vertices is obtained from a k-tree H with n vertices by adding
a vertex and making it adjacent to a k-clique in H.
A partial k-tree is a subgraph1 of a k-tree.
The last concept we define in this section is a generalization of partial k-trees,
called partial k-DAGs. Just as k-trees are a generalization of trees, k-DAGs are a
class of digraphs generated by a generalization of how one might construct a directed,
acyclic graph in a top-down manner.
Definition 7.7 ((Partial) k-DAG). The class of k-DAGs is defined recursively as follows:
• A complete digraph with k vertices is a k-DAG.
• A k-DAG with n + 1 vertices is obtained from a k-DAG H with n vertices by
adding a vertex v and edges satisfying the following:
1 Technically a partial graph is a spanning subgraph, that is, subgraph with the same vertex set. However,
for the results we establish the distinction is not significant.
CHAPTER 7. KELLY-WIDTH
132
– Edges from v to X ⊆ V (H) where |X| ≤ k
– An edge from u ∈ V (H) to v if (u, w) ∈ E(H) for all w ∈ X \ {u}.
A partial k-DAG is a subgraph of a k-DAG.
The second condition on the edges provides a method to add as many edges as
possible going to the new vertex without introducing cycles. Note that if X = ∅, the
antecedent of this condition is true for all u ∈ V (H), so a digraph is a partial 0-DAG
if, and only if, it is a directed acyclic graph.
We also observe that this definition generalizes k-trees, for if the vertices (X) adjacent to the new vertex (v) induce a clique, we will add edges back from X to v,
effectively creating bidirected edges between v and X (and possibly some additional
edges from H \ X to v). The following result shows that k-DAGs generalize the alternative characterization of k-trees we presented initially.
Lemma 7.8. Let G be a k-DAG. Then:
(i) G contains no (k + 2)-clique as a subgraph,
(ii) Any cycle in G with at least three vertices contains a chord, and
(iii) Any bidirected cycle with at least four vertices contains a bidirected chord.
Proof. (i): Let W ⊆ V (G) be a set of k + 2 vertices. Suppose v ∈ W was the last
vertex of W to be added in the construction of G. Since all other vertices of W were
added before v, all edges from v to W were added as part of the first condition on the
added edges. Therefore, there must be at most k outgoing edges from v to vertices in
W , and so W cannot be the vertex set of a (k + 2)-clique.
(ii): Let C = (v1 , v2 , . . . , vn ) be a cycle of length n ≥ 3 in G. Without loss of
generality, assume v0 was the last vertex of C to be added in the construction of G.
Since there is an edge from vn to v1 , it follows that there must be an edge from vn to
all successors of v1 added before v1 , in particular to v2 . Thus (vn , v2 ) is a chord of C.
(iii): Let C = (v1 , v2 , . . . , vn ) be a bidirected cycle of length n ≥ 4. Again we
assume v1 was the last vertex of C to be added in the construction of G. From the
proof of (ii), there is an edge (vn , v2 ) ∈ E(G). Since (v1 , vn , . . . , v2 ) is also a cycle,
the same argument implies there is also an edge (v2 , vn ) ∈ E(G). These two edges
make up a bidirected chord of C.
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Lemma 7.8 does not provide an equivalent characterization for k-DAGs because
the given properties are invariant under edge-reversal. We see in Proposition 7.40 that
the class of k-DAGs is not closed under this operation.
7.1.4 Equivalence results
We have introduced three notions that can be used to define the complexity of digraphs,
all of which naturally extend measures for undirected graphs. On undirected graphs, the
three measures are equivalent to each other, and also to tree-width [DKT97]. Our main
result of this section is that the three measures introduced are equivalent on digraphs.
7.1. GAMES, ORDERINGS AND K -DAGS
133
Theorem 7.9. Let G be a digraph. The following are equivalent:
1. k + 1 cops have a robber-monotone winning strategy to capture an inert robber
on G.
2. G has a directed elimination ordering of width ≤ k.
3. G is a partial k-DAG.
Proof. 1 ⇒ 2: Suppose k + 1 cops have a robber-monotone winning strategy σ.
Without loss of generality, we assume that only one cop is placed at a time. Let
(X0 , R0 )(X1 , R1 ) · · · be the (unique) search consistent with σ. For each v ∈ V (G),
let xv = min{i : v ∈ Xi }. Since σ involves placing one cop at a time, for distinct
v, w ∈ V (G), xv 6= xw . Let C = (v1 , v2 , . . . , vn ) be the order defined as: vi C vj if,
and only if, xvj < xvi . For convenience, let Vi = {v1 , . . . , vi }, and xi = xvi for all i.
We observe that from the definition of xi , Vi ∩ Xxi = {vi }.
We claim C has width ≤ k. If this were not the case, there must exist vi such that
|suppC (vi )| ≥ k + 1. As |suppC (vi )| ≥ k + 1 and |Xxi | ≤ k it follows that there
exists vj ∈ suppC (vi ) \ Xxi . From the definition of suppC (vi ), we have vi C vj , so
xj < xi . Furthermore, from Lemma 7.5, vj ∈ ReachG[Vi ∪{vj }] (vi ). Therefore, since
/ Rxj ,
Vi ∩ Xxi −1 ∩ Xxi = ∅ and vi ∈ Xxi it follows that vj ∈ Rxi . But since vj ∈
the robber-monotonicity of σ implies vj ∈
/ Rl for all l ≥ xj , contradicting the fact that
vj ∈ Rxi . Thus there exists no such vi with |suppC (vi )| ≥ k + 1, and C has width
≤ k.
2 ⇒ 3: Let C = (v1 , v2 , . . . , vn ) be a directed elimination ordering of G of
width k. For ease of notation, define Xi := suppC (vi ), and let m = n − k. Let K0
be the complete graph on the vertices {vm+1 , vm+2 , . . . , vn }, and let Kj (j ≥ 1) be
the k-DAG formed by adding vm−j+1 to Kj−1 , and edges from vm−j+1 to Xm−j+1
(together with the other edges added from Kj−1 to vn−k−j+1 in the definition of kC
is a subgraph of Kj . The result then
DAGs). We claim that for all 0 ≤ j ≤ m, Gm−j
follows by taking j = m. We prove our claim by induction on j. For the base case
(j = 0) the result is trivial as Kj is a complete graph. Now assume the result is true for
C
. For simplicity let i = m − j. By the definition
j ≥ 0, and consider the graph Gm−j−1
C
) either:
of directed elimination, for every edge (u, v) ∈ E(Gi−1
(a) vi ∈
/ {u, v},
(b) u = vi , or
(c) v = vi .
In the first case, (u, v) ∈ E(GiC ) and therefore in E(Kj ) ⊆ E(Kj+1 ) by the induction
hypothesis. For the second case, (u, v) is added during the construction of Kj+1 . For
C
, so (u, w) is an edge of GiC
the final case, for any w ∈ Xi , (vi , w) is an edge of Gi−1
(for u 6= w), and therefore of Kj by the induction hypothesis. Thus (u, vi ) is added
C
) ⊆ E(Kj+1 ) as required.
during the construction of Kj+1 , and E(Gi−1
3 ⇒ 1: Let G be a partial k-DAG. Suppose G is a subgraph of the k-DAG, K,
formed from a complete graph on the vertices Xk := {v1 , v2 , . . . , vk }, and then by
adding the vertices vk+1 , vk+2 , . . . , vn . For 1 ≤ i ≤ n−k let Xk+i ⊆ {v1 , . . . , vk+i−1 }
CHAPTER 7. KELLY-WIDTH
134
denote the set of successors of vk+i . That is, when vk+i is added during the construction of K, edges are added from vk+i to each vertex in Xk+i . Note that for all i,
|Xi | ≤ k. We define a (history dependent) strategy σ for the cops inductively as follows. For all R, σ(∅, R) = Xk . If X = Xi for some i,
k ≤ i ≤ n then for all R, σ(X, R) = Xi ∪ {vi }. If X = Xi ∪ {vi } for some i,
k ≤ i < n, then for all R, σ(X, R) = Xi+1 . We claim that this defines a monotone
winning strategy for k + 1 cops. Let Ri = {vj : j > i}, then from the definition of
k-DAGs and the Xi , it is easy to see that the search associated with the strategy is:
(∅, V (G))(Xk , Rk )(Xk+1 , Rk )(Xk+1 ∪ {vk+1 }, Rk+1 ) · · · (Xn ∪ {vn }, ∅).
As Ri ⊇ Ri+1 for all i, the strategy is monotone and winning as required.
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7.2 Kelly-decompositions and Kelly-width
Theorem 7.9 shows that the concepts introduced in the previous section define a sound
measure of digraph complexity which naturally generalizes tree-width. We now turn
to the problem of finding a closely related digraph decomposition. The decomposition we introduce is a partition of the vertices, arranged as a directed acyclic graph,
together with sets of vertices which guard against paths in the graph that do not respect
this arrangement. We have an additional restriction to avoid trivial decompositions:
vertices in the guard sets must appear either to the left or earlier in the decomposition.
Before we present the formal definition, we recall from Definition 6.13, the definition
of guarding.
Definition 7.10 (Kelly-decomposition and Kelly-width). A Kelly-decomposition of a
digraph G is a triple D := (D, B, W) where D is a DAG and B = (Bd )d∈V (D) and
W = (Wd )d∈V (D) are families of subsets of V (G) such that
(K1) B is a partition of V (G),
(K2) for all d ∈ V (D), Wd guards B≥d :=
S
d0 D Dd Bd0 ,
and
(K3) for all d ∈ V (D) there is a linearSorder on its successors d1 , . . . , dp so that for all
1 ≤ i ≤ p, Wdi ⊆ Bd ∪ W
Sd ∪ j<i B≥dj . Similarly, there is a linear order on
the roots such that Wri ⊆ j<i B≥rj .
The width of D is max{|Bd ∪ Wd | : d ∈ V (D)}. The Kelly-width of G is the minimal
width of any of its Kelly-decompositions.
Our main result of this section is that Kelly-decompositions do in fact correspond
with the complexity measure defined at the end of the previous section.
Theorem 7.11. Let G be a digraph. The following are equivalent:
1. k cops have a robber-monotone winning strategy to capture an inert robber on
G.
2. G has Kelly-width ≤ k.
7.2. KELLY-DECOMPOSITIONS AND KELLY-WIDTH
135
Proof. 2 ⇒ 1: Let (D, B, W) a Kelly-decomposition of G of width k. Let T be the
spanning tree of D obtained from the depth-first traversal of D which always chooses
the greatest successor according to the ordering on successors guaranteed by (K3).
Let (t1 , t2 , . . . , tn ) be the order of V (T ) (and hence, V (D)) visited in the depth-first
traversal of T which always chooses the least successor according to the ordering. So
t1 will always be the root of D which is first in the linear order on the roots, t2 will
be the least successor of t1 which is not a descendant of any greater root, or the next
root of D in the ordering if no such successor exists, and so on. We observe that by the
construction of this ordering, every descendant tj of ti in D is either a descendant of
ti in T , or ti and tj have a common ancestor from which ti is a descendant of a lesser
successor than tj . In both cases j ≥ i from the depth-first traversal of T . It follows
that
[
(7.1)
Btj ∩ B≥ti = ∅.
j<i
We now define the strategy. For 1 ≤ i ≤ n, let X2i−1 = Wti and X2i = Wti ∪
Bti . We define a (history dependent) strategy σ inductively as σ(∅, R) = X1 and
σ(Xi , R) = Xi+1 for all R ⊆ V (G). We claim that σ is a robber-monotone winning
strategy for k cops. Let (X0 , R0 ) · · · (X2n , R2n ) be the search associated
S with the
strategy. We show by induction on i that for 0 < i ≤ n, R2i−2 = R2i−1 = j≥i B≥tj .
It follows immediately that the strategy must
S be monotone and winning. Since X1 =
Wt1 = ∅,Swe have R1 = R0 = V (G) = j≥1 B≥tj . Now let us assume R2i−2 =
R2i−1 = j≥i B≥tj for some i ≥ 1. From (K2), we observe that ReachG\Wti (Bti ) ⊆
B≥ti ⊆ R2i−1 . Thus
R2i = R2i−1 ∪ ReachG\(X2i−1 ∩X2i ) (R2i−1 ∩ X2i ) \ X2i
= R2i−1 ∪ ReachG\Wti (Bti ) \ Bti
S
= j≥i B≥tj \ Bti
S
= j≥i+1 B≥tj (from Equation 7.1).
Since Wd ∩ B≥d = ∅ for allSd ∈ V (D), it follows from (K3) and the construction of
the ordering
S that Wti+1S⊆ j≤i Btj . Therefore, from Equation 7.1, we have R2i ∩
Wti+1 ⊆ j>i B≥tj ∩ j≤i Btj = ∅. Hence,
R2i+1
= R2i ∪ ReachG\(X2i ∩X2i+1 ) (R2i ∩ X2i+1 ) \ X2i+1
= (R2i ∪ ∅) \ Wti+1
= R2i ,
completing the inductive step.
1 ⇒ 2: It follows from Theorem 7.9 that it suffices to show that if G has a directed elimination ordering of width k − 1 then G has Kelly-width ≤ k. Let C =
(v1 , v2 , . . . , vn ) be a directed elimination ordering of G of width k − 1. We define
(D, B, W) as follows. V (D) := V (G). For all d ∈ V (D) let Bd := {d} and
Wd := suppC (d) and define B := (Bd )d∈V (D) and W := (Wd )d∈V (D) . Towards
defining the edge relation of D, let d ∈ V (D) be a node. For convenience we write Gd
for the induced subgraph G[{w : w C d} ∪ {d}]. Let C1 , . . . , Cp be the strongly con-
CHAPTER 7. KELLY-WIDTH
136
nected components of Gd \d. Let d1 , . . . , dp be the C-maximal elements of C1 , . . . , Cp ,
respectively. We put an edge (d, di ) between d and di if di is reachable from d in Gd
and there is no dj with di C dj C d such that dj is reachable from d in Gd and di is
reachable from dj in Gd \ d.
We claim that (D, B, W) is a Kelly-decomposition of width ≤ k. Clearly, D is
a DAG, as all the edges in E(D) are oriented following the ordering C. Further, the
width of the decomposition is clearly at most one more than the width of C.
To establish (K2), we first show the following claim.
Claim. For all d ∈ V (D), ReachGd (d) = B≥d .
Proof of claim. We first show by induction on the index i of d in C that ReachGd (d) ⊆
B≥d . For i = 1 there is nothing to show. Suppose the claim has been proven for all
j < i. Let v ∈ ReachGd (d). Let C1 , . . . , Cm be the strongly connected components of
Gd \ d. Without loss of generality we assume that v ∈ C1 . Let s be the C-maximal
element of C1 and let d0 be the C-maximal element such that
• d0 is the C-maximal element of some Ci
• there is a directed path from d to d0 in Gd
• there is a directed path from d0 to s in Gd \ d.
By construction, there is an edge (d, d0 ) ∈ E(D). If d0 = v, or in fact if d0 is the
C-maximal element of C1 , then there is nothing more to show. Otherwise, if d0 and v
are not in the same strongly connected component of Gd \ d, then s, and hence v, must
be reachable from d0 in Gd0 . For, by construction, s is reachable from d0 in Gd \ d and
d0 is the C-maximal element reachable from d in Gd and from which s can be reached
in Gd \ d. Thus, if s was not reachable from d0 in Gd0 then the only path from d0 to s in
Gd \ d must include an element w C d such that d0 C w, contradicting the maximality
of d0 . Hence, v is reachable from d0 in Gd0 and therefore, by induction hypothesis,
v ∈ B≥t0 ⊆ B≥t .
A simple induction on the height of the nodes in D establishes the converse.
a
It remains to show that for all d ∈ V (D) there is a linear ordering @ of the
successors d satisfying the ordering condition required by the definition of Kellydecompositions. For successors v 6= v 0 of d define v @ v 0 if v 0 C v, that is, @ is
the inverse ordering of C.
Let d1 , . . . , dm be the successors of d ordered by @. We claim that for all i ∈
{1, . . . , m},
[
B≥dj .
Wdi ⊆ Bd ∪ Wd ∪
j<i
If v ∈ Bd there is nothing to show. If d C v then v ∈ Wd as di C d is reachable from
d and therefore Wdi ∩ {x : d E x} = suppC (di ) ∩ {x : d E x} ⊆ suppC (d) ∩ {x :
d E x} = Wd ∩ {x : d E x}. Finally, suppose v C d. But then, v ∈ B≥d and hence
/ B≥di and di C v.
v ∈ B≥dj for some 1 ≤ j ≤ m. By definition of support sets, v ∈
But then, v ∈
/ B≥dj for all j A i, that is, j C i, as then dj C v and by construction,
w C dj for all w ∈ B≥dj . Hence, v ∈ B≥dl for some dl B di . This completes the proof
of the theorem.
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7.3. ALGORITHMIC ASPECTS OF KELLY-WIDTH
137
The proof of Theorem 7.11 is constructive in that given an elimination ordering of
width k − 1 we construct a Kelly-decomposition of width k, and conversely. In fact,
the proofs establish a slightly stronger statement.
Corollary 7.12. Every digraph G of Kelly-width k has a Kelly-decomposition D =
(D, B, W) of width k such that for all d ∈ V (D):
• |Bd | = 1,
• Wd is the minimal set which guards B≥d , and
• Every vertex v ∈ B≥d is reachable in G \ Wd from the unique w ∈ Bd .
Further, if G is strongly connected, then D has only one root.
We call such a decomposition special.
Just as with the cops and visible robber game, it is easy to see that the cops and inert
robber game satisfies the properties introduced in Section 5.4. The characterization of
Kelly-width by such graph searching games implies that Kelly-width is well behaved
under important structural relations. The proofs of the following results are similar to
those presented in Section 6.3.
Lemma 7.13. Let (D, B, W) be a Kelly-decomposition of a digraph G, and let G 0 be
a subgraph of G. (D, B|G 0 , W|G 0 ) where B|G 0 := (Bd ∩ V (G 0 ))d∈V (D) and W|G 0 :=
(Wd ∩ V (G 0 ))d∈V (D) is a Kelly-decomposition of G 0 .
Corollary 7.14. Let G and G 0 be directed graphs such that G 0 is a subgraph of G. Then
Kelly-width(G 0 ) ≤ Kelly-width(G).
Lemma 7.15. Let G be a directed graph and Kn the complete graph on n vertices.
Kelly-width(G • Kn ) = n · Kelly-width(G).
Lemma 7.16. Let G be a directed union of the digraphs G1 and G2 . Then
Kelly-width(G) = max{Kelly-width(G1 ), Kelly-width(G2 )}.
We observe that from this last result it follows that the Kelly-width of a directed
graph is the maximum Kelly-width of all its strongly connected components.
7.3 Algorithmic aspects of Kelly-width
7.3.1 Computing Kelly-decompositions
In this section we mention several algorithms for computing Kelly-width and Kellydecompositions. The proofs of Theorems 7.9 and 7.11 show that Kelly-decompositions
can efficiently be constructed from directed elimination orderings or monotone winning
strategies, so we concern ourselves with the problem of finding any of the equivalent
characterizations.
138
CHAPTER 7. KELLY-WIDTH
In a recent paper Bodlaender et al. [BFK+ 06] study exact algorithms for computing the (undirected) tree-width of a graph. Their algorithms are based on dynamic programming to compute an elimination ordering of the graph. The algorithms translate
easily to directed elimination orderings and can therefore be used to compute Kellywidth, giving us the following theorem:
Theorem 7.17. The Kelly-width of a graph with n vertices and m edges can be determined in
• O (n + m) · 2n time and O n · 2n space, or
• O (n + m) · 4n time and O(n2 ) space.
Proof. The algorithms we require for these bounds are presented as Algorithm 7.1
and Algorithm 7.2 respectively. Lemmas 7.18 and 7.20 prove that these algorithms are
correct, and Lemmas 7.19 and 7.21 establish the running times and space requirements.
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Algorithm 7.1 K ELLY- WIDTH -DP(G)
let KW (∅) = 0
for k = 1 to |V (G)| do
for each S ∈ [V (G)]k do
for each v ∈ S do
Compute suppS (v) := Nout (ReachS (v)) ∪ {v}
let KW (S) = minv∈S max{KW (S \ {v}), |suppS (v)|}
return KW (V (G))
Lemma 7.18. For any digraph G, K ELLY- WIDTH -DP(G) outputs the Kelly-width of
G.
Proof. We observe that for a directed elimination ordering C = (v1 , . . . , vn ), suppC (vi )
is not dependent on the order of the vertices {v1 , . . . , vi−1 }. The algorithm uses this
observation to reduce the number of possible orderings which need to be considered
from n! to 2n . It is easily seen that |suppS (v)| is v together with the support set of
v in any directed elimination ordering where v is preceded by some ordering of the
remaining elements of S. Thus max{KW (S \ {v}), |suppS (v)|} is one more than the
minimal width of a partial directed elimination ordering on S where v is the last vertex
eliminated. It follows that KW (S) returns one more than the minimal width of a partial directed elimination ordering on S, and thus KW (V (G)) returns the Kelly-width
of G.
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Lemma 7.19. Let G be a digraphwith n vertices and m edges. K ELLY- WIDTH -DP(G)
requires at most O (n + m) · 2n time and O n · 2n space.
Proof. For a set S ⊆ V (G) and a vertex v ∈ V (G), it is readily seen that ReachS (v) can
be computed with a depth-first search from v. Since such a search can be executed in
7.3. ALGORITHMIC ASPECTS OF KELLY-WIDTH
139
time O(n + m) [CLR96], it follows that suppS (v) can be computed in time O(n + m).
The innermost for loop is executed once for each S ⊆ V (G), and loops |S| times. So
if each value for KW (S) is stored as it is computed so that its value
P can be found in
constant time, the total running time for the algorithm is O(n+m) S⊆V (G) O(|S|) =
O (n + m) · 2n .
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Algorithm 7.2 K ELLY- WIDTH -R EC(G, L, S)
if S = {v} for some v then
return suppL (v)
let Opt = ∞
for each S 0 ⊆ S with |S 0 | = b|S|/2c do
Compute w1 = K ELLY- WIDTH -R EC(G, L, S 0 )
0
0
Compute w2 =K ELLY- WIDTH -R EC
(G, L ∪ S , S \ S )
let Opt = min Opt, max{w1 , w2 }
return Opt
Lemma 7.20. For any digraph G, K ELLY- WIDTH -R EC(G, ∅, V (G)) outputs the Kellywidth of G.
Proof. We prove by induction on |S| that K ELLY- WIDTH -R EC(G, L, S) returns one
more than the minimal width of a partial directed elimination ordering on L ∪ S where
the first |L| vertices are elements of L. From our observations regarding suppL (v) in
the proof of Lemma 7.18, we see this is true for |S| = 1. Now suppose it is true for
|S| ≤ s, we show that it is true for all S with |S| ≤ 2s. Consider a single execution
of the for loop. Since |S 0 | = b|S|/2c, it follows by the induction hypothesis that w1 is
one more than the minimal width of a partial directed elimination ordering on L ∪ S 0
where the first |L| elements are from L and w2 is one more than the minimal width of
a partial directed elimination ordering on L ∪ S where the first |L| + |S 0 | elements are
from L ∪ S 0 . Thus, the maximum of w1 and w2 is one more than the minimal width
of a partial directed elimination ordering on L ∪ S where the first |L| elements are
from L, and the next |S 0 | elements are from S 0 . Opt stores the minimum of all these
maxima, over all subsets S 0 with |S 0 | = b|S|/2c. As the minimal width of a partial
directed elimination ordering of L ∪ S where the first |L| elements are from L must be
the minimal width of a partial directed elimination ordering of L ∪ S where the first
|L| elements are from L and the next b|S|/2c elements are from S 0 for some S 0 ⊆ S,
it follows that Opt stores the required value. Thus K ELLY- WIDTH -R EC(G, ∅, V (G))
returns the Kelly-width of G.
u
t
Lemma 7.21. Let G be adigraph with n vertices and m edges. K ELLY- WIDTH -R EC(G, ∅, V (G))
runs in O (n + m) · 4n time and O(n2 ) space.
Proof. Let T (s) be the time required to compute K ELLY- WIDTH -R EC
(G, L, S) when
|S| = s. We prove by induction on s that T (s) = O (n + m) · 4s time. If s = 1,
as we saw in Lemma 7.21, suppL (v) can be computed in O(n + m) time,
so the
s
assumption holds for this case. For s > 1, the algorithm runs in time 2 s/2
T (s/2).
140
CHAPTER 7. KELLY-WIDTH
Using asymptotic approximations of Catalan numbers [GKP98], 2n
∈ O(4n ), so
n
s/2
s
T (s) = O(4 )T (s/2) = O (n + m) · 4 . The space requirement follows from
the observation that at each stage of the recursion we need O(n) space to store the
current subset S 0 of S and the values we have computed. Since the recursion tree has
maximum height dlog |S|e ≤ n, we obtain the space bound of O(n2 ).
u
t
For a given k, the problem whether a digraph G has Kelly-width ≤ k is decided
in exponential time with the above algorithms. As the minimization problem is NPcomplete (it generalizes the NP-complete problem of deciding the tree-width of an
undirected graph), we cannot expect polynomial time algorithms to exist. However,
the exact complexity of determining if a digraph has Kelly-width ≤ k for fixed k is
currently unknown. Clearly a digraph has Kelly-width equal to 1 if, and only if, it
is acyclic, and recently Meister, Telle and Vatshelle [MTV07] exhibited a polynomial
time algorithm for determining if a digraph has Kelly-width 2. So for k ≤ 2 the
problem can be solved in polynomial time. For k > 2 it is an open problem.
Open problem 7.22. For a fixed k > 2, what is the complexity of the following problem: Given a digraph G does G have Kelly-width ≤ k?
It seems plausible that, as in the case of DAG-width, studying strategies in the inert
robber game will lead to a polynomial time algorithm when k is fixed.
7.3.2 Algorithms on graphs of small Kelly-width
In this section we present algorithmic applications of Kelly-decompositions, including
a general scheme that can be used to construct algorithms based on a decomposition.
We assume that a Kelly-decomposition (or even an elimination ordering) has been provided or pre-computed. We give an example algorithm based on this to compute the
winner of a parity game, which runs in polynomial time on graphs of bounded Kellywidth. As the algorithm is similar to the algorithm of the previous chapter, we outline
the major difference between the two.
Dynamic programming algorithms using Kelly-decompositions follow a pattern
similar to algorithms that use tree-decompositions and DAG-decompositions. Starting
with a special Kelly-decomposition (D, B, W) and then working bottom up to compute
S for each node d ∈ V (D) a data set containing information on the set B≥d :=
d0 d Bd . The general pattern is described in Algorithm 7.3.
We illustrate this pattern by briefly presenting an algorithm for computing the winner of a parity game. The full algorithm can be found in [HK07]. The algorithm
is similar to the algorithm based on DAG-decompositions, however the separation of
guard sets in Kelly-decompositions makes the presentation more straightforward. As
with DAG-decompositions, we define a Kelly-decomposition of an arena A as a Kellydecomposition of the underlying directed graph (V (A), E(A)).
Theorem 7.23. For any k, given a parity game (A, χ) and a Kelly-decomposition of
A of width ≤ k, determining if Player 0 has a winning strategy from vI (A) can be
computed in polynomial time.
7.3. ALGORITHMIC ASPECTS OF KELLY-WIDTH
141
Algorithm 7.3 Dynamic programming using a Kelly-decomposition
Given a special Kelly-decomposition (D, B, W):
Leaves: Compute the data set for Bd for all leaves d.
Combine: If d ∈ V (D) is an inner node with successors d1 , . . . , dm ordered by the
ordering guaranteed by the Kelly-decomposition (we observe that such an ordering can be computed easily with a greedy algorithm), S
combine the data sets
computed for B≥d1 , . . . , B≥dm to a data set for the union m
i=1 B≥di .
Update: Update the data set computed in the previous step so that the new vertex u
with Bd = {u} is taken into account. Usually, the vertex u will have been part
of at least some guard sets Wdi .
S
Expand: Finally, expand the data set to include guards in Wd \ i Wdi and also paths
starting at u.
To prove the theorem, we first need some preparation. For the rest of this section fix
a parity game (A, χ) where χ : V (A) → P. We assume that the maximal out-degree of
any vertex in V (A) is 2. Using the inert robber game, it is straightforward to show that
the graph resulting from the modification described in Theorem 2.59 has Kelly-width
at most one more than the original graph.
We recall from the proof of Theorem 6.34 the definitions of resultσ (U, v) and
R ESULT(U, v) for a (not necessarily memoryless) strategy σ for Player 0, a subset
of vertices U ⊆ V (A) and a vertex v ∈ V (A). We show how, for a fixed k and given a
special Kelly-decomposition (D, B, W) of A of width k, to compute R ESULT(B≥d , v)
for each d ∈ V (D) and v ∈ B≥d in polynomial time. As with Theorem 6.34 we
observe that as B≥d has at most k guards (Wd ), |R ESULT(B≥d , v)| ≤ (n + 1)k + 2.
The dynamic programming algorithm can then be presented as follows.
Leaves: It follows with the same argument as the Leaves step in the proof of Theorem 6.34, that for any leaf d ∈ V (D), and vertex v ∈ Bd the set R ESULT(Bd , v)
can be computed in constant time.
Combine: Let d be an inner node of D with successors d1 , . . . , dm ordered
S according
to the ordering guaranteed
by
(K3).
For
1
≤
i
≤
m,
let
B
:=
i
j≤i B≥di and
S
let B := Bm = 1≤i≤m B≥di . We aim to compute the set R ESULT(B, u) for
each u ∈ B. We observe that if i < j and u ∈ B≥di then every path from
u to a vertex v ∈ B≥dj \ B≥di must go through Wd . Hence, if u ∈ B≥di
then R ESULT(B, u) = R ESULT(Bi , u). We compute for each i ≤ m and u ∈
Bi the set R ESULT(Bi , u) by induction on i. For i = 1, R ESULT(B1 , u) =
R ESULT(d1 , u). Let i > 1 and let u ∈ Bi \ Bi−1 .
To compute R ESULT(Bi , u), we do the following. Let r = resultσ (B≥di , u) ∈
R ESULT(B≥di , u) be a set of results against a strategy σ for Player 0. The result
set r gives us the set of vertices v ∈ Wdi to which Player 1 can force the play
against σ and also the best priority he can achieve in doing so. Now, if v ∈
CHAPTER 7. KELLY-WIDTH
142
Wdi ∩ Bi−1 is a guard contained in Bi−1 then once the play has reached v it can
never return to Bi \ Bi−1 and continues in Bi−1 until it reaches a vertex in Wd .
Hence, once the play has reached v, we can determine the results of possible
strategies in Bi−1 from R ESULT(Bi−1 , v).
This suggests the following algorithm for computing R ESULT(Bi , u). For each
r ∈ R ESULT(B≥di , u) we compute a set Rr of sets as follows. Let R :=
{(w, p) ∈ r : w ∈ Wdi \ Wd } be the set of outcomes in r for plays which end in
vertices in Bi−1 . Let (w1 , p1 ), . . . , (ws , ps ) list the elements of R. For each tuple ρ = (r1 , . . . , rs ) with rj ∈ R ESULT(Bi−1 , wj ) Let Rρ be defined as follows.
For each (v, p) ∈ r\R add (v, p) to Rρ . If (v, q) ∈ rj add (v, max{pj , q}) to Rρ .
Then, add the set Rρ to Rr . Then, R ESULT(Bi , u) contains for each Rρ ∈ Rr
the set of E-minimal pairs in Rρ .
Update and Expand: We now consider how to update the data structure to take account of paths that include vertices entering B≥d . The argument is similar to the
Expand step of the proof of Theorem 6.34, so we refer the reader there for the
details.
We observe that each step of the above algorithm, and hence the entire algorithm, runs
in polynomial time. This completes the outline of the proof of Theorem 7.23.
7.3.3 Asymmetric matrix factorization
We saw in Section 7.1.2 that the idea of vertex elimination was motivated by the practical application of solving systems of linear equations. Such systems are more commonly represented as matrix equations: M x = b, with the goal being to find a solution
for the n×1 vector of variables, x, given an m×n matrix M , and an m×1 vector b. A
straightforward solution to such an equation is to find M −1 , the inverse of M , to obtain
x = M −1 b, however a more common approach is to factorize M in such a way that
solutions may be easily computed. Cholesky decompositions and LU-factorizations
are two such examples of this. If M is an m × n matrix, an LU-factorization (or LUdecomposition) of M is an m × m lower triangular matrix L and an m × n upper
triangular matrix U such that M = LU . If, in addition M is symmetric and positive
definite, then there is an LU-factorization of M where U = LT . Such a decomposition is called a Cholesky decomposition. When a matrix has an LU-factorization we
can solve the equation M x = b as follows: first we use forward substitution to solve
Ly = b, and then backward substitution to solve U x = y.
The elimination process we described in Section 7.1.2, also known as Gaussian
elimination, is one of the most common methods for computing an LU-factorization
or a Cholesky decomposition. More precisely, Gaussian elimination is the process
of transforming a matrix into an upper triangular matrix via row operations: adding a
multiple of one row to another (including itself), or interchanging two rows (also known
as pivoting). The resulting upper triangular matrix is the U factor of a LU-factorization,
and the row operations can be represented by a sequence of transformation matrices, the
product of which form the L factor of the LU-factorization. If the original matrix was
symmetric and positive definite, this process will generate a Cholesky decomposition.
7.3. ALGORITHMIC ASPECTS OF KELLY-WIDTH
143
Since Gaussian elimination can be used to compute LU-factorizations and Cholesky
decompositions, it is not surprising that elimination orderings and two associated structures we introduce here, elimination trees and elimination DAGs, are useful for investigating the complexity of computing these matrix decompositions. We first define the
particular relationship between graphs and matrices that we are interested in.
Definition 7.24. Let M = (aij ) be a square n × n matrix. We define GM as the
directed graph with V (GM ) = {v1 , . . . , vn }, and for i 6= j, (vi , vj ) ∈ E(GM ) if, and
only if, aij 6= 0. We also define the elimination ordering CM as CM := (v1 , . . . , vn ).
When M is a symmetric matrix, we view GM as an undirected graph rather than a
bidirected graph.
One structure that is particularly useful for analysing symmetric matrix factorization is the elimination tree.
Definition 7.25 (Elimination tree). Let G be an undirected graph, and C an elimination
ordering for G. The elimination tree defined by C is a pair (T , λ) where T is a rooted
tree and λ : V (T ) → V (G) is a bijection such that if s ∈ V (T ) is the parent of
t ∈ V (T ), then λ(s) = minC suppC (λ(t)) .
Liu [Liu90] observed that elimination trees can be used to investigate many aspects of Cholesky decompositions, for example the row and column structure of the
Cholesky factors can be extracted directly from an elimination tree. Another observation, from Bodlaender et al. [BGHK95], is that the height of an elimination tree gives
the parallel time required to compute a Cholesky decomposition of a symmetric matrix
using Gaussian elimination.
In [GL93], Gilbert and Liu introduced a generalization of elimination trees, called
elimination DAGs, which can be similarly used to analyse factorizations in the asymmetric case. We recall that a transitive reduction of a directed graph is a minimal graph
with the same transitive closure and we observe that an acyclic graph has a unique
transitive closure.
Definition 7.26 (Upper and Lower elimination DAGs [GL93]). Let M be a square
matrix that can be decomposed as M = LU without pivoting. The upper (lower)
elimination DAG is the transitive reduction of the directed graph GU (GL respectively).
Gilbert and Liu [GL93] observed that elimination DAGs enjoy many properties
similar to elimination trees. For instance, they are an efficient storage scheme for sparse
matrices, and an upper and lower pair of elimination DAGs are sufficient to capture the
path structure of a graph: if there is a directed path from u to v in the graph, then there
is a vertex w such that there is a path from u to w in the upper elimination DAG, and a
path from w to v in the lower elimination DAG. They also showed that when the matrix
is symmetric, the upper elimination DAG is isomorphic to the elimination tree, as is
the lower elimination DAG when its edges are reversed.
The Kelly-decomposition constructed in the proof of Theorem 7.11 captures the
upper and lower elimination DAGs in a very direct manner.
Theorem 7.27. Let M be a square matrix that can be decomposed as M = LU without
pivoting. Let (D, B, W) be the Kelly-decomposition of GM obtained by applying the
proof of Theorem 7.11 with elimination order CM . Then
144
CHAPTER 7. KELLY-WIDTH
(a) (D, B) is isomorphic to the lower elimination DAG, and
(b) GU = (V (GM ), {(v, w) : w ∈ Wv }), thus the upper elimination DAG is isomorphic to the transitive reduction of the relation {(v, w) : w ∈ Wv }.
Proof. For v ∈ V (GM ), let Xv = {v}∪{w ∈ V (GM ) : wCM v}. First, from Theorem
1 of [RT78]:
(E(GL ))T C = {(v, w) : w CM v, and there is a path from v to w in GM [Xv ]},
where RT C denotes the transitive closure of R. We observe that in the construction of
the Kelly-decomposition, E(D) is the transitive reduction of the right-hand side. Since,
by construction, elements of B are singletons, we can view B as a bijection between
V (D) and V (G), and the first result follows. Secondly, from Theorem 4.6 of [GL93],
we have
E(GU ) = {(v, w) : v CM w, and there is a v 0 ∈ ReachXv (v) with (v 0 , w) ∈ E(GM )}.
The second result then follows from Lemma 7.5, which shows that {(v, w) : w ∈
u
t
Wv } = {(v, w) : w ∈ suppCM (v)} is equivalent to the right-hand side.
We can use the results of [GL93] to make the following observation when we construct Kelly-decompositions on undirected graphs.
Corollary 7.28. Let G be an undirected graph, C an elimination order on G and
(D, B, W) the Kelly-decomposition of G (considered as a bidirected graph) obtained
by applying the proof of Theorem 7.11 with elimination order C. Then D is a tree,
and more precisely, (Dop , B) is isomorphic to the elimination tree associated with the
(undirected) elimination order C.
7.4 Comparing Kelly-width and DAG-width
In this section we use graph searching games to compare Kelly-width to DAG-width
and directed tree-width. In the undirected case, all the games we consider require the
same number of searchers, however we show that in the directed case there are graphs
on which all three measures differ by an arbitrary amount. We show that Kelly-width
bounds directed tree-width within a constant factor, but the converse fails as there are
classes of graphs of bounded directed tree-width and unbounded Kelly-width. We also
provide evidence to suggest that Kelly-width and DAG-width are within a constant
factor of each other.
We recall from Definition 6.1 the cops and robber game used to characterize DAGwidth. For convenience, we will refer to this as the visible robber game. In Example 5.2.1 we discussed another cops and robber game that partially characterizes directed tree-width: the strongly connected visible robber game. The following theorem
summarizes Theorems 6.15 and Lemma 5.41:
Theorem 7.29. Let G be a digraph.
7.4. COMPARING KELLY-WIDTH AND DAG-WIDTH
145
1. G has DAG-width k if, and only if, k cops have a monotone winning strategy in
the visible robber game on G.
2. G has directed tree-width ≤ 3k + 1 or k cops do not have a winning strategy in
the strongly connected visible robber game on G.
For the undirected case, the following proposition sums up results from [DKT97]
and [ST93].
Proposition 7.30. On any undirected graph G, the following are equivalent
1. k cops have a winning strategy in the visible robber game.
2. k cops have a robber-monotone and cop-monotone winning strategy in the visible robber game.
3. k cops have a winning strategy in the inert robber game.
4. k cops have a robber monotone winning strategy in the inert robber game.
5. The tree-width of G is at most k − 1.
It follows from these results that Kelly-width is a generalization of tree-width in
the following sense.
←
→
Corollary 7.31. Let G be an undirected graph. G has tree-width k if, and only if, G
has Kelly-width k.
On general directed graphs, the situation is more complicated. As we saw in Theorem 6.11, monotonicity is not sufficient for the visible robber game. Kreutzer and
Ordyniak [KO07] have also recently shown that monotonicity is not sufficient for the
inert robber game.
Theorem 7.32 ([KO07]). For any m ∈ N, there exists a graph for which 6m cops
can capture an invisible, inert robber but 7m cops are required to do so with a robbermonotone strategy.
Of course, as with Theorem 6.11, this does not preclude the possibility that the
number of cops required for monotonicity is bounded by some factor of the number of
cops required with any strategy.
Open problem 7.33. Does there exist a function f : ω → ω such that for all digraphs
G, if k cops can capture an inert robber on G then f (k) cops can capture the robber
with a robber-monotone strategy?
Before we compare Kelly-width with directed tree-width and DAG-width, we first
observe that Proposition 7.2 allows us to compare Kelly-width and directed path-width.
As we mentioned previously, Barát [Bar05] observed that the directed path-width of a
digraph was one less than the minimum number of cops required to capture an invisible
robber with a cop-monotone strategy. Thus, using the observation that a cop-monotone
strategy in the cops and inert robber game is also robber-monotone, and the example
from Proposition 6.39, we obtain the following relationship between Kelly-width and
directed path-width.
CHAPTER 7. KELLY-WIDTH
146
Proposition 7.34.
(i) If a directed graph G has directed path-width k, it has Kelly-width at most k + 1.
(ii) There exists a family of graphs with arbitrarily large directed path-width and
Kelly-width 2.
Our next comparison result shows that a robber-monotone winning strategy in the
inert robber game can be translated to a (not necessarily monotone) winning strategy
in the visible robber game.
Theorem 7.35. Let G be a directed graph. If k cops can catch an inert robber with a
robber-monotone strategy on G, then 2k − 1 cops can catch a visible robber on G.
Proof. Suppose k cops have a robber-monotone winning strategy in the inert robber
game on a digraph G. By Theorem 7.9 this implies that there is a directed elimination
ordering C on G of width ≤ k − 1. We use the elimination ordering to describe the
winning strategy of 2k − 1 cops against a visible robber, thereby establishing the result.
The cops are split into two groups, k cops called the blockers and k − 1 cops called
the chasers. Similarly, the cop moves are split in two phases, a blocking move and a
chasing phase.
In the first move, k cops are placed on the k highest elements with respect to C.
These cops form the set of blockers. Let the robber choose some element v. This
concludes the first (blocking) move. We observe:
If u is the C-smallest vertex occupied by a blocker, then every directed path from v to a vertex greater than u has at least one vertex
occupied by a cop.
(∗)
This invariant is maintained by the blocking cops during the play. Now suppose after r
rounds have been played, the robber occupies vertex v and the blockers occupy vertices
in X so that the invariant (∗) is preserved. Let u be the C-smallest element in X and let
C1 , . . . , Cs be the set of strongly connected components of G[{u0 : u0 C u}]. Further,
let @ be a linear ordering on C := {C1 , . . . , Cs } so that Ci @ Cj if, and only if, the
C-maximal element in Ci is C-smaller than the C-maximal element of Cj . Now the
cops move as follows. Let C ∈ C be the component such that v ∈ C and let w ∈ C be
the C-maximal element in C. The cops place the k − 1 cops not currently on the graph
on suppC (w). These cops are the chasers. As the chasers approach, the robber has
two options. Either he stays within C or he escapes to a vertex in a different strongly
connected component C 0 . If the robber runs to a vertex x ∈ C or x ∈ C 0 for some
C 0 @ C then after the chasers land on S := suppC (w) there is no path from x to a
node u such that u B u0 for the C-minimal vertex u0 in S. Hence, the chasers become
blockers and the chasing phase is completed. Otherwise, if the robber escapes to a
C 0 with C @ C 0 , then the chasers repeat the procedure and move to suppC (w0 ) for
the C-maximal element in C 0 . However, as the robber always escapes to a @-larger
strongly connected component and also can not bypass the blockers, this chasing phase
must end after finitely many steps with the robber being on a vertex v ∈ C for some
component C and the chasers being on suppC (w) for the C-maximal element in C.
At this point the chasers become blockers. One of the old blockers is now placed on
7.4. COMPARING KELLY-WIDTH AND DAG-WIDTH
147
v1
•
 ???

? v
v3 
v4
/•
/• 2
•/ JJ
/
t
t
// JJ
/
t // JJJJ /// ttt t
//
JJ tt/
// tttJtJJJ/// J tt
•
•
v5
v6
Figure 7.1: Graph G showing the difference between DAG-width and inert robber game
w and all others are removed from the board. The cop on w makes sure that in each
such step the robber space shrinks by at least one vertex. By construction, the invariant
in (∗) is maintained. Further, as the robber space shrinks by at least one after every
chasing-phase, the robber is eventually caught by the cops.
u
t
An immediate consequence of this is that the Kelly-width of a graph bounds the
directed tree-width of the graph.
Corollary 7.36. Let G be a directed graph with Kelly-width k. Then G has directed
tree-width ≤ 6k − 2.
Since it is not known whether the number of cops required for a winning strategy in
the visible robber game bounds the number of cops required for a monotone winning
strategy, we cannot obtain a similar bound for DAG-width. We can, however, ask
whether we can improve the bound. That is, assuming that k cops have a robbermonotone winning strategy against an invisible, inert robber can we define a winning
strategy for less than 2k − 1 cops in the visible robber game? Although it might be
possible to improve the result, the next theorem shows that we cannot do better than
with 43 k cops.
Theorem 7.37. For every m ∈ N, there is a graph such that 3m cops have a robbermonotone winning strategy in the inert robber game but no fewer than 4m cops can
catch a mobile visible robber.
Proof. Consider the graph G in Figure 7.1. We show that on G, 3 cops do not have a
(non-monotone winning) strategy to catch a visible robber, however
4 cops do. Con
sider the partition of V (G), H = {v1 , v2 , v4 }, {v3 }, {v5 }, {v6 } . The strategy for the
robber against 3 cops is to move to any element of H which is not occupied by a cop.
As long as the robber moves to one of {v1 , v4 } when the cops occupy {v3 , v5 , v6 }, it
will always be possible for him to move to such an element when the cops move. However 4 cops can capture a visible robber with a monotone strategy by occupying the
following sequence of sets of vertices: {v3 , v4 , v5 , v6 }, {v2 , v3 , v5 , v6 }, {v1 , v2 , v3 }.
On the other hand, 3 cops suffice to capture an invisible, inert robber with a robbermonotone strategy by occupying the following sequence of sets of vertices: {v4 , v5 , v6 },
CHAPTER 7. KELLY-WIDTH
148
{v3 , v5 , v6 }, {v2 , v5 , v6 }, {v2 , v3 }, {v1 , v2 , v3 }. The result follows by taking the lexicographic product of this graph with the complete graph on m vertices.
u
t
Since 4 cops can capture a visible robber with a monotone strategy on the graph in
the previous proof, we have the following:
Corollary 7.38. For all m ∈ N there are graphs of DAG-width 4m and Kelly-width
3m.
Despite this
4
3
bound, for graphs of small Kelly-width we can do better.
Theorem 7.39. For k = 1 or 2, if G has Kelly-width k, G has DAG-width k.
Proof. If G has Kelly-width 1, then it must be acyclic, as all guard sets are empty.
Thus it has DAG-width 1. If G has Kelly-width 2, then it has an elimination ordering
C = (v1 , v2 , . . . , vn ) of width 1. A cop-monotone strategy for two cops against a
visible robber is as follows. Initially, let i = n and place one cop on vi . At this point,
the robber is restricted to {v1 , . . . , vi−1 }. Let j < i be the maximal index such that
the robber can reach vj . Place a cop on vj . After the cop has landed, we claim that
the robber is unable to reach both vi and vj . For otherwise, let r be the maximal index
such that the robber can reach vr (with cops on vi and vj ) and from vr can reach vi
(with a cop on vj ) and vj (with a cop on vi ). By the maximality of j, r < j. Let s > r
be the first index greater than r which occurs on a path from vr to vi that does not go
through vj , and t > r be the first index greater than r which occurs on a path from vr
to vj that does not go through vi . Then from the maximality of r, s 6= t. Furthermore,
{vs , vt } ⊆ suppC (r), so |suppC (vr )| > 1, contradicting the width of the ordering. So
we can remove the cop from whichever vertex the robber can no longer reach without
changing the robber space, and either the robber is now restricted to {v1 , . . . , vj } or
the maximal index which the robber can reach is smaller. Clearly, this is a monotone
winning strategy for two cops.
u
t
We now turn to the converse problem, what can be said about the Kelly-width of
graphs given their directed tree-width or DAG-width?
Firstly we observe the following analogue of Proposition 6.7 for Kelly-width.
Proposition 7.40. For any j, k with 2 ≤ j < k, there exists a graph Tkj such that
Kelly-width(Tkj ) = j and Kelly-width((Tkj )op ) = k.
Proof. Consider the graph Tkj from Proposition 6.7. In the proof of Proposition 6.7,
the strategies described for the cops and the robber are also winning strategies in the
inert robber game.2
u
t
It follows, using the same argument of Proposition 6.37 that there are families of
graphs of bounded directed tree-width and unbounded Kelly-width.
Corollary 7.41. There exist families of digraphs with directed tree-width 2 and unbounded Kelly-width.
2 Indeed,
the winning strategy for the robber is winning even if the robber is visible and inert.
7.4. COMPARING KELLY-WIDTH AND DAG-WIDTH
149
Our final result is a step towards relating Kelly-width to DAG-width by showing
how to translate a monotone strategy in the visible robber game to a (not necessarily
monotone) strategy in the inert robber game.
Theorem 7.42. If G has DAG-width ≤ k, then k cops have a winning strategy in the
inert robber game.
Proof. Given a DAG-decomposition (D, X ) of G of width k, the strategy for k cops
against an invisible, inert robber is to follow a depth-first search on the decomposition.
More precisely, we assume the decomposition has a single root r, and we have an
empty stack of nodes of D.
1. Initially, place the cops on Xr and push r onto the stack.
2. At this point we assume d is on the top of the stack and the cops are on Xd . We
next “process” the successors of d in turn. To process a successor d0 of d, we
remove all cops not on Xd ∩ Xd0 , place cops on Xd0 , push d0 onto the stack, and
return to step 2. Note that a node may be processed more than once.
3. Once all the successors of a node have been processed, we pop the node off the
stack and if the stack is non-empty, return to step 2.
Because the depth-first search covers all nodes of the DAG and hence all vertices of
the graph are eventually occupied by a cop, the robber will be forced to move at some
point. Due to the guarding condition for DAG-decompositions, when the robber is
forced to move this strategy will always force the robber into a smaller region and
eventually capture him.
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Again we observe that it is unknown if, in the inert robber game, the number of
cops required to capture the robber with a robber-monotone strategy is bounded by the
number of cops required to capture him with any strategy. So this result does not allow
us to directly compare Kelly-width and DAG-width. However, we strongly believe that
the number of cops required for monotone strategies is bounded in both the inert robber
game and the visible robber game, giving us the following conjecture:
Conjecture 7.43. The Kelly-width and DAG-width of a digraph lie within constant
factors of one another.
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CHAPTER 7. KELLY-WIDTH
Chapter 8
Havens, Brambles and Minors
for Directed Connectivity
In this chapter we present some preliminary work towards a structure theory for directed graphs based on directed connectivity. The aim of such a structure theory would
be to obtain generalizations of some of the significant results for undirected graphs, for
example finding a directed analogue of the Graph Minor Theorem. However, as we
show, even determining some of the basic building blocks of such a structure theory
leads to some interesting open problems. We work on the assumption that DAG-width,
Kelly-width and the non-monotone versions of their cops and robber games are all approximately the same and can therefore be used to measure the directed connectivity of
a digraph. Then, using the premise that DAG-width or Kelly-width measures the complexity of a graph, we consider the following two questions: What structural features
are present in directed graphs which are “complex”?; and what relation on directed
graphs indicates “structural simplification”?
As we observed with Theorem 4.7 the existence of a bramble or a haven in an
undirected graph indicates that the tree-width is not going to be small. Similarly, Theorems 4.7 and 4.11 show that the existence of the natural generalizations of havens and
brambles imply that the directed tree-width is not going to be small. So in order to
address the first question, we consider generalizations of havens and brambles which
correspond to DAG-width and Kelly-width. Although we are unable to show full equivalence as with Theorems 4.7 and 4.11, we can show, via cops and robber games, that
they do provide obstructions for DAG-width and Kelly-width. That is, their existence
in a graph places restrictions on the DAG-width or Kelly-width of that graph.
Towards finding a relation which indicates structural simplification, we consider the
problem of extending the minor relation to directed graphs. As we mentioned in Chapter 4, the minor relation is an important relation in the structural theory of undirected
graphs as it indicates whether one graph is structurally more simple than another. So
having a minor relation for directed graphs is the cornerstone of any digraph structure
theory. We argue that the existing definitions in the literature of minors for directed
graphs are not sufficient, in the sense that a structure theory based on them would not
151
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CHAPTER 8. HAVENS, BRAMBLES AND MINORS
be able to produce similar results to those of undirected graph structure theory. While
it may be the case that there is no appropriate relation for directed graphs, we provide
some examples which may take the investigation further.
8.1 Havens and brambles
The aim of this section is to define various structural properties which may lead to a
minimax theorem for DAG-width and Kelly-width, similar to Theorem 4.7. To achieve
this, we introduce some generalizations of havens and brambles and show how they
relate to DAG-width and Kelly-width. We recall from Chapter 4, the definitions and
theorem that we wish to generalize:
Definition 4.5 (Haven). Let G be an undirected graph and k ∈ N. A haven of order k
in G is a function β : [V (G)]<k → P(V (G)) such that for all X ⊆ V (G) with |X| < k:
(H1) β(X) is a non-empty connected component of G \ X, and
(H2) If Y ⊆ X, then β(Y ) ⊇ β(X).
Definition 4.6 (Bramble). Let G be an undirected graph. A bramble in G is a set B of
connected subsets of V (G) such that for all pairs B, B 0 ∈ B either B ∩ B 0 6= ∅, or
there exists {u, v} ∈ E(G) with u ∈ B and v ∈ B 0 . The width of a bramble B is the
minimum size of a set which has a non-empty intersection with every element of B.
Theorem 4.7 ([ST93]). Let G be an undirected graph. The following are equivalent:
1. G has tree-width ≥ k − 1
2. G has a haven of order k.
3. G has a bramble of width k.
We saw with Theorems 4.11 and 4.12, that the natural extension of these definitions to directed graphs – replacing “connected components” with “strongly connected
components” – results in structural properties closely related to directed tree-width. In
this section we introduce some less obvious extensions that are closer to DAG-width
and Kelly-width. One of the major obstacles to finding such definitions, and the reason why the extensions we consider are less obvious is that the definitions have to be
dependent on edge direction. That is, a bramble or haven of a graph should not necessarily be a bramble or haven of the graph obtained by reversing the direction of the
edges. The above definitions of haven and bramble do not have obvious extensions
which satisfy this property, however the definitions we introduce next are dependent
on edge direction.
Definition 8.1 (D-Haven). Let G be a directed graph and k ∈ N. A D-haven of order k
in G is a function β : [V (G)]<k → P(V (G)) such that for all X ⊆ V (G) with |X| < k:
(DH1) β(X) is a non-empty subset of V (G \ X), and
(DH2) If Y ⊆ X then β(Y ) ⊇ β(X) and ∀y ∈ β(Y ), β(X) ∩ Reachβ(Y ) (y) 6= ∅.
8.1. HAVENS AND BRAMBLES
153
As suggested by the nomenclature, and as we observed in Chapter 5, on undirected
graphs havens describe winning strategies for the robber in the cops and visible robber game. That is, when the cops are on X, β(X) suggests the locations the robber
should occupy to defeat the cops. The analogous result for the game on directed graphs
suggests that D-havens are the “correct” extension of havens for DAG-width. More
precisely,
Proposition 8.2. Let G be a directed graph. The robber can defeat k cops in the visible
cops and robber game on G if, and only if, G has a D-haven of order k + 1.
Proof. If G has a D-haven β of order k + 1, then the strategy for the robber is to
remain in β(X) whenever the cops are on X. The D-haven axioms guarantee that this
is always possible. More precisely, we define the following strategy for the robber:
ρ(X, X 0 , R) = ReachG\X 0 (r0 ) for some r0 ∈ β(X 0 ) ∩ ReachG\(X∩X 0 ) (r). We observe
that at every position (X, r), r ∈ β(X) and show that this implies that such a choice is
always possible. Since X ⊇ X 0 ∩X, it follows from (DH2) that r ∈ β(X ∩X 0 ). Then,
since X 0 ⊇ X ∩X 0 , β(X 0 )∩Reachβ(X∩X 0 ) (r) 6= ∅, so ρ is well defined. Finally, since
ρ(X, X 0 , r) ∈ ReachG\(X∩X 0 ) (r) by definition, ρ is a valid strategy for the robber in
the cops and visible robber game.
For the converse, suppose the robber has a winning strategy, ρ, against k cops.
Define, for X ∈ [V (G)]≤k ,
[
β(X) := {R : the robber wins from (X, R) playing ρ}.
We show that β is a D-haven of order k + 1. We observe that ρ(∅, X, V (G)) ⊆ β(X),
so as ρ is a winning strategy, β(X) is non-empty for all X ∈ [V (G)]≤k . Thus (DH1)
holds. For (DH2) we observe from the definition of the cops and visible robber game
that if the robber can win from (X, R) then he can win from (Y, R) for all Y ⊆ X.
Thus, if Y ⊆ X, then β(Y ) ⊇ β(X).
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Immediately from this result and Lemmas 6.18, 6.20 and 6.21,we observe that
D-havens behave as we expect under subgraphs, lexicographic product, and directed
union. Also as a consequence of Proposition 8.2, the existence of a D-haven in a digraph imposes a restriction on the DAG-width of the graph.
Corollary 8.3. Let G be a digraph. If G has a D-haven of width k then the DAG-width
of G is at least k.
Since a D-haven corresponds to a winning strategy for the robber against any cop
strategy and DAG-width corresponds to monotone winning strategies, the converse of
Corollary 8.3 is equivalent to the monotonicity question for the cops and visible robber
game: if k cops have a winning strategy, do k cops have a monotone winning strategy?
As there are graphs where more cops are required to capture the robber with a monotone strategy [KO07], we know that this does not hold in general. However, a result
similar to Theorem 4.11 would provide a solution to the more general monotonicity
problem posed in Open Problem 6.12.
Obdržálek [Obd06] observed that the relaxation of connected components in (H1)
to subsets in (DH1) is necessary if we require havens to correspond to strategies for the
robber. More precisely, let us say that a D-haven, β, is connected if it also satisfies:
154
CHAPTER 8. HAVENS, BRAMBLES AND MINORS
v1
•
v2 x
•
8•
v3
•
v4
Figure 8.1: Graph to show that D-havens may be disconnected
(DH10 ) β(X) is a non-empty weakly connected component of G \ X.
Proposition 8.4 ([Obd06]). There exists a directed graph G such that the robber can
defeat 2 cops in the cops and visible robber game on G, but there is no connected
D-haven of order 2 in G.
The graph that illustrates this result is shown in Figure 8.1. It is difficult to define
a notion of haven that corresponds to the inert robber game for two reasons. First,
because the motility of the robber is dependent on the move of the cops, there may be
a number of “responses” to a given cop position in this game. So having a function
defined only for sets of cop locations is not going to be sufficient. Secondly, as we
observed in Chapters 5 and 7, the cops and robber game with an invisible robber is
essentially a single player game. Thus there is only one strategy for the robber and it is
either winning or it is not. So having a function which dynamically suggests a strategy
for the robber is not going to be particularly interesting. A more practical approach
would be to identify the structural features which are present when the strategy for the
robber is winning. This leads us to the problem of extending the definition of brambles.
Before we introduce the extension of brambles we are interested in, we need to
introduce the concept of initial and terminal components.
Definition 8.5 (Initial and Terminal Component). Let G be a directed graph, and H
a strongly connected component of G. H is an initial component if it is closed under
predecessors. That is, if v ∈ V (G) with (v, w) ∈ E(G) for some w ∈ V (H), then
v ∈ V (H). H is a terminal component if it is closed under successors. That is, if
v ∈ V (G) with (w, v) ∈ E(G) for some w ∈ V (H), then v ∈ V (H).
We denote by Init(G) the set of all vertices in initial components, and Term(G) the
set of all vertices in terminal components. For a subset of vertices B ⊆ V (G) we write
Init(B) and Term(B) for Init(G[B]) and Term(G[B]) respectively when G is clear from
the context.
Another way to view initial and terminal components are as the roots and leaves
(respectively) of the block graph of G: the directed acyclic graph with the strongly
connected components of G as vertices and an edge (C, C 0 ) if there is an edge in G
from some vertex in C to some vertex in C 0 . With this interpretation it is straightforward to show that initial and terminal components are well-behaved with respect to the
structural relations for directed graphs we consider important.
Lemma 8.6. Let G, G 0 and G 00 be non-empty directed graphs and C ⊆ G an initial
(terminal) component of G.
1. If G 0 is a subgraph of G with C ∩ V (G 0 ) 6= ∅ then there is an initial (terminal)
component C 0 ⊆ G 0 such that C 0 ⊆ C.
8.1. HAVENS AND BRAMBLES
155
2. If G is a directed union of G 0 and G 00 then C is either an initial (terminal) component of G1 or an initial (terminal) component of G2 .
3. If G 0 is a directed union of G and G 00 (directed union of G 00 and G) then C is an
initial (terminal) component of G 0 .
4. If either |C| ≥ 2 or G 0 is strongly connected, then C • G 0 is an initial (terminal)
component of G • G 0 .
5. If G = G 0 •G 00 then π1 (C) = {v : (v, w) ∈ C} is an initial (terminal) component
of G 0 .
Definition 8.7 (Initial bramble). Let G be a directed graph. An initial bramble in G is
a set B of subsets of V (G) such that for all pairs B, B 0 ∈ B and for all x ∈ Init(B),
there exists y ∈ Init(B 0 ) such that y ∈ ReachB∪Init(B 0 ) (x).
Definition 8.8 (Terminal bramble). Let G be a directed graph. A terminal bramble in G
is a set B of subsets of V (G) such that for all pairs B, B 0 ∈ B and for all x ∈ Term(B),
there exists y ∈ Term(B 0 ) such that y ∈ ReachTerm(B)∪B 0 (x).
Definition 8.9 (Bramble width). Let G be a directed graph and B an initial or terminal
bramble in G. The width of B is the size of the smallest hitting set of B. That is, the
size of the minimal X ⊆ V (G) such that X ∩ B 6= ∅ for all B ∈ B.
Although it would appear that initial and terminal brambles are similar entities,
we show that there are graphs where the smallest width of an initial bramble differs
from the smallest width of a terminal bramble. It might also seem that, since an initial
component of a graph G is a terminal component of the graph G op obtained by reversing
the direction of the edges of G, that an initial bramble in G is a terminal bramble in G op .
However, the ordering of the quantifiers in each of the definitions means that this is
not necessarily the case: an initial bramble in G is, in G op , a set of subsets such that
for all pairs B, B 0 and all x ∈ Term(G op [B]), there exists y ∈ Term(G op [B 0 ]) such
that x ∈ ReachG op [Term(B 0 )∪B] (y). Before we show how initial and terminal brambles
differ, we show how they correspond to DAG-width and Kelly-width, and establish
some robustness results.
Lemma 8.10. Let G be a directed graph.
1. If G has an initial bramble of width k then the robber can defeat k − 1 cops in
the cops and visible robber game.
2. If G has a terminal bramble of width k then the robber can defeat k − 1 cops in
the cops and inert robber game.
Proof. 1: Suppose G has an initial bramble B of width k. Then, for any set X with
|X| ≤ k − 1 there exists BX ∈ B such that BX ∩ X = ∅. The strategy for the robber is
to be on some vertex in Init(BX ) whenever the cops are located on X. It is clear from
the definition of an initial bramble that such a move is always possible. As the robber
is able to do this forever, it follows that this is a winning strategy for the visible robber
against k − 1 cops.
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CHAPTER 8. HAVENS, BRAMBLES AND MINORS
2: Now suppose G has a terminal bramble B of width k. Again, for any set X
with |X| ≤ k − 1 there exists BX ∈ B such that BX ∩ X = ∅. The “strategy” for
the robber is, when he can move and when the cops are on X, to move to the first
element of a strongly connected component of Term(BX ) that will be occupied by the
cops. More precisely, we show that after every cop move, there exists B ∈ B such
that Term(B) is contaminated. Clearly this is true at the beginning, as every vertex is
contaminated. Now suppose the cops are moving from X to X 0 and for some B ∈ B
and some terminal component C of G[B], X ∩ C = ∅ and there exists a contaminated
vertex v ∈ X 0 ∩ C. As BX 0 ∩ X 0 = ∅, and C is a terminal component, the path in
Term(B) ∪ BX 0 from v to some w ∈ Term(BX 0 ) is cop-free. Thus BX 0 is now an
element of B such that Term(BX 0 ) is contaminated.
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An immediate corollary from the game characterizations of DAG-width and Kellywidth is that initial and terminal brambles provide obstructions for DAG-width and
Kelly-width.
Corollary 8.11. Let G be a directed graph.
1. If G has an initial bramble of width k then G has DAG-width ≥ k.
2. If G has a terminal bramble of width k then G has Kelly-width ≥ k.
Unfortunately, it is not known whether the converse to Lemma 8.10 holds.
It is relatively straightforward to show that brambles behave manner to the cops
and robber games under various graph operations. For example a bramble of a graph
is a bramble of any supergraph, and the width of a bramble increases by an appropriate factor under lexicographic products. This strongly suggests that the converse of
Lemma 8.10 does hold.
Conjecture 8.12. Let G be a directed graph.
1. If the robber can defeat k − 1 cops in the cops and visible robber game on G then
G has an initial bramble of width k.
2. If the robber can defeat k − 1 cops in the cops and inert robber game on G then
G has a terminal bramble of width k.
We observe that since monotonicity is not sufficient in either cops and robber
game [KO07], we know that the converse of Corollary 8.11 does not hold. However,
as with D-havens, a result along the lines of Theorem 4.12 would resolve Open Problems 6.12 and 7.33.
We conclude this section by combining these results with some results from Chapter 7 to show that initial brambles and terminal brambles are different.
Proposition 8.13. For all m ∈ N, there exists a directed graph with an initial bramble
of width 4m but no terminal bramble of width ≥ 3m + 1.
Proof. Consider the graph G in Figure 7.1. As we observed in the proof of Theorem 7.37, 3 cops suffice to capture
an inert robber on G. Wealso showed that G has
an initial bramble of width 4: {v1 , v2 , v4 }, {v3 }, {v5 }, {v6 } . The result follows by
taking the lexicographic product of this graph with Km , the complete digraph on m
vertices.
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8.2. DIRECTED MINORS
157
8.2 Directed minors
In this section we investigate the problem of finding a relation on directed graphs which
represents structural simplification. Such relations are ubiquitous throughout mathematics, for example in algebra or model theory homomorphisms describe structural
simplifications, and in geometry or topology homeomorphisms are the key structural
relations. Graphs can be viewed both as relational structures and as topological complexes, so there are well-defined notions of graph homomorphisms and graph homeomorphisms. However, for undirected graphs at least, the minor relation is arguably the
most suitable relation for comparing fundamental graph structural properties such as
connectivity and cyclicity. Intuitively, a graph G is a minor of a graph H if G can be
embedded in H modulo connected sets. That is, if we consider connected sets in H as
“vertices”, then G is a subgraph of this “graph”. More precisely,
Definition 8.14 (Minor). Let G and H be undirected graphs. G is a minor of H, written
G ≤ H, if there exists a function ξ : V (G) → P(V (H)) which maps distinct vertices
to disjoint sets such that:
• For all v ∈ V (G), H[ξ(v)] is a connected graph, and
• For all {v, w} ∈ E(G) there exists {v 0 , w0 } ∈ E(H) such that v 0 ∈ ξ(v) and
w0 ∈ ξ(w).
So why is the minor relation a good indicator of structural simplification? As we
observed above, there are well-defined notions of graph homomorphisms and graph
homeomorphisms. A homomorphism preserves relational structure and a homeomorphism preserves topological shape, so injective homomorphisms or subgraph homeomorphisms would seem to be reasonable indicators of structural simplification. However, the minor relation subsumes these. We see from the definition that the minor
relation can be considered a generalization of injective graph homomorphisms: G is a
minor of H if there is a homomorphic-like injective map from V (G) to connected sets
of H. Presently we will also show that if G is homeomorphic to a subgraph of H then
G is a minor of H. So the minor relation can be seen as a generalization of both relational and topological structure simplification. We now turn to the problem of finding
an extension of the minor relation to directed graphs which enjoys similar properties.
The definition of a minor has two obvious extensions to directed graphs: either map
vertices to weakly connected sets or map vertices to strongly connected sets. However,
as we argue below, neither of these truly reflect the notion of structural simplification
that complexity measures like directed tree-width, DAG-width and Kelly-width suggest. In the remainder of this section we identify the characteristics of the minor relation that make it useful and we introduce several definitions of digraph minor relations
and compare them against these criteria. First we show how we can view the minor
relation operationally, and how this implies that the minor relation is a generalization
of subgraph homeomorphism.
Definition 8.15 (Edge contraction). Let G be a graph, and e = (v, w) ∈ E(G). The
graph G 0 obtained from G by contracting e is defined as:
• V (G 0 ) = V (G) \ {v},
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CHAPTER 8. HAVENS, BRAMBLES AND MINORS
• E(G 0 ) = E(G) ∪ {(u, w) : (u, v) ∈ E(G)} ∪ {(w, u) : (v, u) ∈ E(G)} \
{(u, v), (v, u) : u ∈ V (G)}.
The following result follows easily from the definitions and is often used as an
alternative definition of the minor relation.
Lemma 8.16. Let G and H be undirected graphs. The following are equivalent:
1. G is a minor of H,
2. G is isomorphic to a subgraph of a graph obtained by contracting edges of H,
and
3. G is isomorphic to a graph obtained by contracting edges of a subgraph of H.
Proof. 1 ⇒ 2: Suppose G is a minor of H and let ξ : V (G) → P(V (H)) be the
function witnessing this. Let H0 be the graph obtained from H by contracting, for each
v ∈ V (G) the edges in H[ξ(v)]. Now ξ can be viewed as an injective mapping from
V (G) to V (H0 ) such that for each {v, w} ∈ E(G), {ξ(v), ξ(w)} ∈ E(H0 ). That is, ξ
is an embedding of G in H0 , so G is isomorphic to a subgraph of H0 , a graph resulting
from contracting edges of H.
2 ⇔ 3: Let us view the subgraph relation as the operation of deleting edges and
isolated vertices. That is, G is a subgraph of H if G can be obtained by deleting edges
and isolated vertices of H. We observe that edge and isolated vertex deletion and edge
contraction commute, that is we obtain the same graph independent of the order of
the operations. Thus if we perform all edge contractions first and then all deletions we
obtain the same graph by performing all deletions first followed by all edge contractions
and vice versa. Thus any subgraph of a graph obtained by contracting edges is a graph
obtained by contracting edges of a subgraph and conversely.
3 ⇒ 1: Suppose G is isomorphic to a graph obtained by contracting edges of
H0 where H0 is a subgraph of H. For convenience, we will assume that G is a graph
obtained by contracting edges of H0 . For each v ∈ V (G) define ξ(v) as the set of
vertices w ∈ V (H0 ) such that there is a path from w to v consisting of edges which
are contracted to obtain G. From the definition of ξ, H[ξ(v)] = H0 [ξ(v)] is connected.
Now suppose {v, w} ∈ E(G). It follows from the definition of edge contractions that
there exists {v 0 , w0 } ∈ E(H0 ) such that there are paths from v 0 to v and from w0 to w
consisting of edges which are contracted to obtain G. That is v 0 ∈ ξ(v) and w0 ∈ ξ(w).
As V (H0 ) ⊆ V (H), h is a function from V (G) to P(V (H)), so G is a minor of H. u
t
Indeed, as subgraphs and edge contractions are well-defined for directed graphs,
this lemma suggests the following natural definition of a minor relation on directed
graphs.
Definition 8.17 (Minor for digraphs). Let G and H be directed graphs. G is a minor of
H, G ≤ H, if G is isomorphic to a graph obtained from H by a sequence of edge and
isolated vertex deletions and edge contractions.
It is clear from Lemma 8.16 that this definition is equivalent to the minor relation
on the underlying undirected graphs, hence the notation. That is,
8.2. DIRECTED MINORS
159
Proposition 8.18. Let G and H be digraphs. G ≤ H if, and only if, G ≤ H.
We also observe that the ≤-minor relation corresponds to the weakly connected
“natural” generalization of the minor relation.
Proposition 8.19. Let G and H be digraphs. G ≤ H if, and only if, there exists a
function ξ : V (G) → P(V (H)) such that:
• if v 6= w then ξ(v) is disjoint from ξ(w),
• for all v ∈ V (G), H[ξ(v)] is a weakly connected graph, and
• for all (v, w) ∈ E(G) there exists (v 0 , w0 ) ∈ E(H) such that v 0 ∈ ξ(v) and
w0 ∈ ξ(w).
These observations show that the minor relation has a straightforward extension to
directed graphs. However, just the simple extension of tree-width to directed graphs
is not an ideal measure of complexity, we argue below that this definition is not restrictive enough to be a suitable relation for structural simplification for digraphs. In
particular a minor of an acyclic digraph need not be acyclic, which goes against our
tenet that acyclic graphs are structurally the least complex graphs. However, all the
minor relations we introduce in the Section 8.2.2 are restrictions of this relation.
Lemma 8.16 also demonstrates how minors can be seen as a generalization of subgraph homeomorphisms. First we recall the definition of a subgraph homeomorphism.
Definition 8.20 (Subgraph homeomorphism). Let G and H be (directed) graphs. We
say G is homeomorphic to a subgraph of H if there is an injective function η : V (G) →
V (H) and a mapping p from edges of G to pairwise internal-vertex-disjoint paths in H
such that for e = (v, w) ∈ E(G), p(e) is a (directed) path from η(v) to η(w).
Lemma 8.21. Let G and H be undirected graphs. If G is homeomorphic to a subgraph
of H then G is a minor of H.
Proof. We observe that if G is homeomorphic to a subgraph of H, then G is isomorphic
to a graph obtained from a subgraph of H by repeatedly replacing vertices of degree 2
with an edge joining its neighbours. But this operation can also be viewed as contracting edges that have at least one endpoint with degree 2. Therefore G is isomorphic to
a graph obtained by contracting edges of a subgraph of H, so by Lemma 8.16, G is a
minor of H.
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8.2.1 What makes a good minor relation?
We now consider the properties we expect a reasonable definition of a minor relation
for directed graphs to satisfy. First and foremost, the relation should respect digraph
complexity. That is, if G is a minor of H then G should not be more structurally complex than H. But which notion of digraph complexity should we use? As we mentioned
at the start of the chapter we are primarily interested in a relation corresponding to directed connectivity, so DAG-width or Kelly-width or their associated cops and robber
games would be suitable. However, there is also no known appropriate relation for
CHAPTER 8. HAVENS, BRAMBLES AND MINORS
160
strong connectivity, so we also consider directed tree-width. In Section 8.2.3 we consider various graph properties that are preserved under the operation “taking a minor”
and use these to identify unsuitable candidates.
The second property we are interested in is being able to obtain generalizations of
theorems concerning the minor relation. In particular, we are concerned with trying to
extend two results: the Graph Minor Theorem, which asserts that the minor relation is
a well-quasi order, and the algorithmic result that for a fixed graph H, determining if
H is a minor of G can be decided in cubic time. The latter result implies that any class
characterized by a finite set of excluded minors can be decided in polynomial time,
and the former implies that any minor-closed property can be characterized by a finite
set of excluded minors. Although we show that many of our defined relations fail to
satisfy this property, the investigation raises some interesting questions.
Our final requirement for a reasonable notion of a minor relation for directed graphs
is that it should be an extension of the minor relation for undirected graphs. In partic←
→
ular, if G and H are undirected graphs such that G is a minor of H then G should be
←
→
a minor of H . Furthermore, it should also generalize subgraph homeomorphisms (for
directed graphs). That is, if we replace internal-vertex-disjoint paths with single edges
we should obtain a minor of the original graph. Although many of our defined relations do satisfy both these requirements, some interesting relations do not, including
the strongly connected “natural” generalization of the minor relation and two relations
which occur in the literature: the butterfly minor relation and the topological minor
relation.
8.2.2 Directed minor relations
In this section we define several minor relations for digraphs. We adopt the operational
definition of minor implied by Lemma 8.16 and generate variations by considering
different restrictions on the edge contraction operation. For the results we establish, it
is convenient to consider two types of edge contraction operation: one which contracts
a single edge, and one which contracts multiple edges simultaneously. We call the
first kind edge contractions and the second set contractions. We observe that when a
sequence of edge contractions are performed, it does not matter in which order they are
performed, the resulting graphs are all isomorphic. Thus to “simultaneously” contract
a set of edges, we can contract them individually in some arbitrary order. We now
define the edge and set contractions we use to define our minor relations.
Definition 8.22. Let G be a directed graph and e = (u, v) ∈ E(G).
• We say e can be topologically contracted if either
– u has in-degree 1 and out-degree 1, or
– v has in-degree 1 and out-degree 1.
• We say e can be butterfly contracted if either
– u has out-degree 1, or
– v has in-degree 1.
8.2. DIRECTED MINORS
161
• We say e can be D-contracted unless either
– there is a directed path from u to v edge disjoint from (u, v), or
– there exists two vertex disjoint cycles C1 , C2 , each with at least two vertices, such that u ∈ C1 and v ∈ C2 .
Before we introduce the set contractions, we observe that the above definitions of
edge contractions are ordered from most restrictive to least restrictive. That is,
Lemma 8.23. Let G be a directed graph and e = (u, v) ∈ E(G). If e can be topologically contracted then e can be butterfly contracted, and if e can be butterfly contracted
then e can be D-contracted.
Proof. If e can be topologically contracted then clearly e can be butterfly contracted.
Now suppose e can be butterfly contracted. If u has out-degree 1 then e is the only
outgoing edge from u so there is no path from u to v which is edge disjoint from e and
there is no cycle which contains u and does not contain v. Thus e can be D-contracted.
Otherwise v has in-degree 1 and e is the only incoming edge to v. Again, there can be
no path from u to v which is edge disjoint from e and there is no cycle which contains
v and does not contain u. So e can be D-contracted.
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t
Definition 8.24. Let G be a directed graph and E ⊆ E(G).
• If E = {(u, v), (v, u)} then the simultaneous contraction of E is an anti-parallel
contraction.
• If G[E] is a strongly connected graph, then the simultaneous contraction of E is
a strong contraction.
Clearly these definitions are also ordered from most restrictive to least restrictive.
We include the result for completeness.
Lemma 8.25. Let G be a directed graph and E ⊆ E(G). An anti-parallel contraction
of E is a strong contraction of E.
We now combine these edge and set contractions with the subgraph relation to
obtain a number of minor relations.
Definition 8.26 (Subgraph minor). Let G and H be directed graphs. G is a subgraph
minor of H, G b H, if G is isomorphic to a graph obtained from H by a sequence
of edge and isolated vertex deletions. G is an anti-parallel subgraph minor of H,
G bAP H, if G is isomorphic to a graph obtained from H by a sequence of edge and
isolated vertex deletions and anti-parallel contractions. G is a strong subgraph minor
of H, G bS H, if G is isomorphic to a graph obtained from H by a sequence of edge
and isolated vertex deletions and strong contractions.
Definition 8.27 (Topological minor). Let G and H be directed graphs. G is a topological minor of H, G a H, if G is isomorphic to a graph obtained from H by a sequence
of edge and isolated vertex deletions and topological contractions. G is an anti-parallel
topological minor of H, G aAP H, if G is isomorphic to a graph obtained from H
162
CHAPTER 8. HAVENS, BRAMBLES AND MINORS
by a sequence of edge and isolated vertex deletions, and anti-parallel and topological
contractions. G is a strong topological minor of H, G aS H, if G is isomorphic to a
graph obtained from H by a sequence of edge and isolated vertex deletions, and strong
and topological contractions.
Definition 8.28 (Butterfly minor). Let G and H be directed graphs. G is a butterfly
minor of H, G H, if G is isomorphic to a graph obtained from H by a sequence
of edge and isolated vertex deletions and butterfly contractions. G is an anti-parallel
butterfly minor of H, G AP H, if G is isomorphic to a graph obtained from H
by a sequence of edge and isolated vertex deletions, and anti-parallel and butterfly
contractions. G is a strong butterfly minor of H, G S H, if G is isomorphic to a
graph obtained from H by a sequence of edge and isolated vertex deletions, and strong
and butterfly contractions.
Definition 8.29 (D-minor). Let G and H be directed graphs. G is a D-minor of H,
GEH, if G is isomorphic to a graph obtained from H by a sequence of edge and isolated
vertex deletions and D-contractions. G is an anti-parallel D-minor of H, G EAP H, if
G is isomorphic to a graph obtained from H by a sequence of edge and isolated vertex
deletions, and anti-parallel and D-contractions. G is a strong D-minor of H, G ES H, if
G is isomorphic to a graph obtained from H by a sequence of edge and isolated vertex
deletions, and strong and D-contractions.
Remark. Unlike the case for the undirected minor relation, the edge and set contractions we have defined here do not commute with edge and vertex deletion: an edge
may not be edge contractible until some other edges have been deleted, and a set of
edges may no longer be set contractible after some edges have been deleted. However,
for our definitions it is the case that the reverse holds: if an edge is edge contractible
before some other edges or vertices have been deleted, then it is still edge contractible
after those deletions, and if a set of edges is set contractible after some deletions then it
is set contractible before those deletions. So we may assume that to obtain a minor we
perform a sequence of set contractions, followed by a sequence of edge and isolated
vertex deletions, followed by a sequence of edge contractions.
Before we establish some results, we define a useful function which captures the
inverse of edge contraction.
Definition 8.30 (Vertex expansion). Let be a minor relation, and let G and H be
directed graphs such that G H. A -vertex expansion of G to H is a function
ξ : V (G) → P(V (H)) defined to be ξ n in the following construction. Let G0 G1 . . . Gn be a sequence of graphs such that G0 = H, Gn = G and Gi+1 is obtained
from Gi by a single edge deletion, vertex deletion, or edge contraction1. For each i ≤ n
define ξ i : V (Gi ) → P(V (H)) as follows. ξ 0 (v) = {v}. If Gi+1 is obtained from Gi by
contracting (u, v) then ξ i+1 (v) = ξ i (v) ∪ ξ i (u) and ξ i+1 (w) = ξ i (w) for all w 6= u, v
(recall that u ∈
/ V (Gi+1 )). Otherwise we let ξ i+1 (w) = ξ i (w) for all w ∈ V (Gi+1 ).
Lemmas 8.23 and 8.25 imply that all the minor relations we have so far defined
can be arranged as in the inclusion diagram of Figure 8.2. Presently we will show that
1 We treat set contractions as sequences of single edge contractions, so G might not necessarily be a
i
minor of Gi+1
8.2. DIRECTED MINORS
163
≤?
??
ES ?
??


S
EAP?
?
??
??






E
AP?
aS ?
??
?? 







bS ?
aAP?
??
?? 


AP
a
b ?
?? 

b
Figure 8.2: Inclusion diagram for the introduced minor relations
each inclusion in Figure 8.2 is strict, however first we need to show that these minor
relations are well-behaved with respect to directed connectivity.
Theorem 8.31. Let G and H be directed graphs, with G ES H. If k cops can capture
a visible robber on H then k cops can capture a visible robber on G.
Proof. As a consequence of Lemma 6.18, it suffices to show that the number of cops
required decreases after either a D-contraction or a strong contraction. Let ξ be a ES vertex expansion from G to H. The idea is that if any of the vertices of ξ(w) is occupied
by a cop, then we occupy w with a cop. It is clear that if G is obtained from H by strong
contractions only, then this describes a winning strategy for the cops as the robber is
more restricted in his movement. So it suffices to consider the case when G is obtained
from H by a single D-contraction of the edge (u, v). In this case, the robber may be
able to reach some vertices in G that he could not reach in H by a directed path through
the contraction of u and v. Let U ⊆ V (H) be the set of vertices w, not including u, for
which there is a path from u to w edge disjoint from (u, v), and let V ⊆ V (H) be the
set of vertices w, not including v, for which there is a path from w to v edge disjoint
from (u, v). We observe that after (u, v) is contracted, the robber is able to move from
vertices in V to vertices in U . We argue that he can only do this once.
Since (u, v) can be D-contracted, U and V are disjoint, as otherwise there would
be a path from u to v edge disjoint from (u, v). Thus, any path from U to V must
include the edge (u, v). For any x ∈ V , suppose there is a directed path in G to some
y ∈ U such that there is a directed path from y to some z ∈ V . Since such a path in
H must go through (u, v), it follows that there is a path from y to u and a path from v
to z. Thus u and y are two distinct vertices in a cycle, as are v and z, contradicting the
assumption that (u, v) could be D-contracted.
The strategy for the cops is now as follows. Play as before, occupying w ∈ V (G)
if some vertex in ξ(w) is occupied. If the robber never moves from V to U , then each
move of the robber can be simulated on H. Otherwise, if the robber does move from
V to U , he can never return to V , so we can discard this part of the graph and continue
playing the winning strategy on the subgraph of H.
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164
CHAPTER 8. HAVENS, BRAMBLES AND MINORS
We now have sufficient tools to demonstrate that each minor relation we have defined is distinct from the others, and that there are no other inclusions other than those
we have already identified.
Theorem 8.32. The inclusion diagram of Figure 8.2 is strict and complete.
Proof. To prove the result, it suffices to show the following six inequations:
(I) a 6⊆ bS
(II) 6⊆ aS
(III) E 6⊆ S
(IV) ≤ 6⊆ ES
(V) bAP 6⊆ E
(VI) bS 6⊆ EAP
Now consider Table 8.1. We show that for each pair of minor relations (, 0 ), G2 G1 but G2 60 G1 . It is easy to see that in each example, G2 G1 . We therefore show
that G2 60 G1 .
(I) – (III): We observe that in each example, the graph G2 has only one less edge
than G1 . It is easily checked that deleting any edge from G1 will not result in the
graph G2 , thus the only possible way for G2 0 G1 is from edge contractions. In (I),
by symmetry any edge will suffice. But no single edge is contractible under strong
contractions, thus G2 6bS G1 . In (II) and (III), to obtain a vertex of degree 4, the only
edge which can be contracted is the vertical edge. However, in (II) both endpoints of
this edge have out-degree 2 and therefore it cannot be topologically contracted, and in
(III) this edge is neither the only outgoing edge of its tail nor the only incoming edge
of its head, thus it cannot be butterfly contracted. In both cases it cannot be contracted
using a strong contraction, thus in (II) G2 6aS G1 , and in (III) G2 6S G1 .
(IV): This follows directly from Theorem 8.31, as G1 is acyclic and G2 is not.
(V): We observe that it is not possible to D-contract any edge in G1 . Thus if G2
is a D-minor of G1 , G2 must be a D-minor of some subgraph of G1 with at least one
edge deleted. However, only two cops are required to capture a robber on G1 with any
edge deleted, whereas three cops are required to capture a robber on G2 . Thus, from
Theorem 8.31, G2 cannot be a D-minor of G1 .
(VI): We observe that it is not possible to D-contract any edge in G1 without first
deleting some edges. As anti-parallel contractions reduce the number of anti-parallel
pairs of edges, we cannot obtain G2 from G1 through anti-parallel contractions alone.
Thus if G2 EAP G1 , to obtain G2 from G1 we must first delete some edges. However,
it is easy to check that after any edge is deleted from G1 , three cops have a winning
strategy to capture a visible robber: intuitively, removing an edge makes one of the
small cycles “weaker” than the others, either by removing one of the edges which
leaves the cycle, or removing one of the edges in the cycle. The strategy for three
cops is then to chase the robber into this weaker cycle, and then use the weakness to
capture him. As four cops are required to capture the robber on G2 , it follows from
Theorem 8.31 that G2 6 EAP G1 .
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8.2. DIRECTED MINORS
A 6⊆ B
(I)
a 6⊆ bS
(II)
6⊆ aS
(III)
E 6⊆ S
(IV)
≤ 6⊆ ES
(V)
bAP 6⊆ E
(VI) bS 6⊆ EAP
165
G1
• o •O
• /•
/•
/•
•
•o
•
•
•
/•
/•
/•
/•
/•
•?
??
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•o
•
•
•
•
•
• 3o E•1
3 1
• 111
11
11
•X1
11
1
11
11
11
/ • MM
•
o^^ qqq
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o
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•11 F•
•0 F•
0 1 •
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11
/•
•
•
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/•
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•?
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/•
/•
•
•
•
•
11
11
•
•
•1
111
•M 11
qqqq MMM1
•
•
Table 8.1: Separating examples of the introduced minor relations
166
CHAPTER 8. HAVENS, BRAMBLES AND MINORS
Remark. Example (IV), which shows that ≤ 6⊆ ES , illustrates that a ≤-minor of an
acyclic graph may not necessarily be acyclic. This supports our earlier claim that ≤
was not restrictive enough to be a reasonable indicator of structural simplification for
directed graphs.
Before we consider some other structural properties which are preserved by the
minor relations we have defined, we show that the relations we have introduced include
other digraph relations that we have already considered. First we show that topological
minors correspond to directed subgraph homeomorphisms.
Proposition 8.33. Let G and H be directed graphs. G a H if, and only if, G is homeomorphic to a subgraph of H.
Proof. Using the proof of Lemma 8.21 we see that if G is homeomorphic to a subgraph
of H then G a H, as the edge contractions used in the proof are all topological contractions. For the converse, suppose G a H. Without loss of generality, we may assume
G is obtained by a sequence of edge and vertex deletions followed by a sequence of
topological contractions. Thus G is obtained from a subgraph H0 of H by a sequence
of topological contractions. Let ξ : V (G) → P(V (H0 )) be a a-vertex expansion. We
show how ξ can be used to define a (directed) subgraph homeomorphism. From the
definition of topological contraction, we observe that for each u ∈ V (G), there is at
most one u0 ∈ ξ(u) with out-degree ≥ 1, as otherwise it would not be possible to contract ξ(u) to a single vertex. This means that intuitively, H0 [ξ(u)] looks like a star with
one central vertex, paths radiating outwards, and a path from u to the central vertex.
We define η : V (G) → V (H) by setting η(u) to be either the vertex in ξ(u) with more
than one successor, or u if there is no such vertex. We observe the following: if the
in-degree of u is greater than 1, then η(u) = u; there is a directed path in ξ(u) from
u to η(u); and there is a directed path in ξ(u) from η(u) to all vertices in ξ(u) with a
successor outside of ξ(u). Now let (u, v) ∈ E(G) be an edge in G. From the definition
of edge contraction, there exists w ∈ ξ(u) such that (w, v) ∈ E(H0 ). From our observations regarding η(u) and ξ(u), it follows that there exists a path from η(u) to v.
Since there is a path from v to η(v), it follows that there is a path from η(u) to η(v). To
show this path is vertex distinct (excluding end-points) from any other, we observe that
for any v 0 6= v such that (u, v 0 ) ∈ E(G), the path from η(u) to v 0 is disjoint (except
for η(u)) to the path from η(u) to v, and if u0 6= u is a predecessor of v in G, then
η(v) = v, so the paths from η(u) to η(v) and from η(u0 ) to η(v) are disjoint.
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t
Now we observe that the strong subset minor relation corresponds to the strongly
connected “natural” generalization of the minor relation.
Proposition 8.34. Let G and H be digraphs. G bS H if, and only if, there exists an
function ξ : V (G) → P(V (H)) which maps distinct vertices to disjoint sets such that:
• for all v ∈ V (G), H[ξ(v)] is a strongly connected graph, and
• for all (v, w) ∈ E(G) there exists (v 0 , w0 ) ∈ E(H) such that v 0 ∈ ξ(v) and
w0 ∈ ξ(w).
Proof. Let ξ be a bS -vertex expansion of G in H. From the definition of strong contraction and edge contraction, it follows that ξ satisfies the requirements.
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8.2. DIRECTED MINORS
167
Finally we observe from Lemma 8.16 that if a minor relation allows anti-parallel
contractions, then on bidirected graphs the relation is equivalent to the minor relation
for undirected graphs.
Proposition 8.35. Let G and H be undirected graphs, and a minor relation such
←
→ ←
→
that ⊇bAP . Then G ≤ H if, and only if G H .
8.2.3 Preservation results
Theorem 8.31 showed that all the minor relations we introduced respect complexity
as defined by directed connectivity. We now consider some other structural properties
that are preserved under the operation of taking a minor. Our first result shows that the
taking of butterfly minors preserves non-reachability, or equivalently, a butterfly minor
vertex expansion preserves reachability.
Proposition 8.36. Let G and H be digraphs such that G S H. Let ξ be a S -vertex
expansion of G in H. Let u, v ∈ V (G). If there is a directed path from u to v then there
exists u0 ∈ ξ(u) and v 0 ∈ ξ(v) such that there is a directed path from u0 to v 0 .
Proof. Clearly if G is a subgraph of H then the result holds, and similarly if G can
be obtained from H by strong contractions. Thus it suffices to assume that G can be
obtained from H by butterfly contractions. Let w ∈ V (G) be a vertex of G Since ξ(w)
butterfly contracts to a single vertex, it follows that there exists a vertex w0 ∈ ξ(w)
such that there is a path to w0 from all vertices in ξ(w) with in-degree greater than
1, and there is a path from w0 to all vertices in ξ(w) with out-degree greater than 1.
Furthermore, there is a path from w to w0 and a path from w0 to all vertices in ξ(w)
with a successor not in ξ(w). If w0 w1 · · · wn is a path in G from u = w0 to v = wn ,
let wi0 be the vertex in ξ(wi ) which satisfies the above observation. It follows from
the definition of edge contraction, that for all i ≥ 0, there is a path in H from wi0 to
0
wi+1
(in ξ(wi ) ∪ ξ(wi+1 )). Thus there exists a path from u0 = w00 to v 0 = wn0 , as
required.
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Example (III) in Table 8.1 shows that Proposition 8.36 does not hold for D-minors.
However, D-minors do preserve a more restrictive structural property: strong connectivity.
Proposition 8.37. Let G and H be digraphs such that G ES H. Let ξ be a ES -vertex
expansion of G in H. Let u, v ∈ V (G). If there are directed paths from u to v and from
v to u then there exists u0 ∈ ξ(u) and v 0 ∈ ξ(v) such that there are directed paths from
u0 to v 0 and from u0 to v 0 .
Proof. As with Proposition 8.36, we observe that we can assume that G can be obtained
from H by D-contractions. For w ∈ V (G), we observe from the definition of Dcontractions that H[ξ(w)] takes the following form: a directed tree, rooted at w, such
that if w1 w2 · · · wn is a path in H with w1 , wn ∈ ξ(w), then wn is an ancestor of w1
in H[ξ(w)]. For if this were not the case, then it would not be possible to D-contract
ξ(w) to a single vertex. The result now follows by expanding the vertices in the cycle
containing u and v in a similar way to Proposition 8.36.
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168
CHAPTER 8. HAVENS, BRAMBLES AND MINORS
8.2.4 Algorithmic results
We now consider the algorithmic aspects of the minor relations we have defined. In
particular, we are concerned with the following decision problem:
(G, )- MINOR
Instance: A directed graph H
Problem: Is G H?
In [RS95], it was shown that for undirected graphs and the standard minor relation,
(G, ≤)- MINOR is solvable in cubic time, so it is worth investigating if any of the minor
relations we have defined enjoy a similar property. Unfortunately, we show that this is
not the case unless NP = P TIME, as the problem is in general NP-complete for most
of the relations we have defined.
Fortune, Hopcroft and Wyllie [FHW80] showed a dichotomy result for the directed
subgraph homeomorphism problem for a fixed pattern graph G . If G is a star, that is
there is a unique source or sink which is the tail or head (respectively) of every edge,
then deciding if a given graph with a given node mapping has a subgraph homeomorphic to G is solvable in polynomial time. Otherwise it is NP-complete. Not surprisingly,
from Proposition 8.33, this result is partly applicable to (G, a)- MINOR. The difference
is that in [FHW80] it is assumed that the node mapping was given. That is, they were
asking if given a node mapping could be extended to a subgraph homeomorphism. The
(G, a)- MINOR problem corresponds to the case when the node mapping is not given.
This case was discussed in [FHW80] where it was observed that firstly the polynomial
time result carries over, as there are at most a polynomial number of node mappings,
and secondly with some additional structure in the pattern graph, the node mapping
required for NP-completeness can be forced to be the only possible node mapping, so
the NP-completeness result holds for a large class of directed graphs (but not quite the
complement of the star graphs). Summarizing their results in the terminology of this
chapter gives us:
Theorem 8.38 ([FHW80]). If G is a directed graph which is a star then (G, a)- MINOR
is solvable in polynomial time.
Theorem 8.39 ([FHW80]). If G is a directed graph with at least four distinct vertices
{v1 , v2 , v3 , v4 } and edges (v1 , v2 ) and (v3 , v4 ) such that for i ≤ 4 the degree of vi
is greater than 3 and different from the degree of vj for j 6= i, then (G, a)- MINOR is
NP-complete.
Corollary 8.40. Let be a minor relation which includes a and let G be a directed
graph which satisfies the requirements of Theorem 8.39. Then (G, )- MINOR is NPcomplete.
Because of the additional structure required in the pattern graph to show NPcompleteness when the node mapping is not specified, we no longer have the dichotomy result. Indeed it is an interesting problem to investigate the complexity of
the problem when G is neither a star nor a directed graph satisfying the requirements of
Theorem 8.39, for example if the maximum degree of any vertex in G is 3. This gives
us the following problem for further investigation.
8.2. DIRECTED MINORS
•
111
11
1
C3 111
•
•
169
•
•
C4
•
•
v • HHH
vv
HH
v
v
•
•)
))
C5
))
•
•
···
Figure 8.3: An infinite anti-chain for the E relation
Open problem 8.41. Characterize G, and the class of graphs H such that (G, )MINOR is NP-complete
8.2.5 Well-quasi order results
We conclude this chapter by showing that only a few of the relations we have introduced can be used to generalize one of the most significant theorems associated with
the minor relation: the Graph Minor Theorem of Robertson and Seymour [RS04]. Recalling the definition of a well-quasi order from Section 1.1.1, the theorem can be stated
as:
Theorem 8.42 (Graph Minor Theorem [RS04]). The minor relation is a well-quasi
order.
In particular this implies that for any infinite set of graphs there is a pair of graphs
such that one is the minor of the other. From this, it follows that any family of graphs
which is closed under the minor relation can be characterized by a finite list of forbidden minors. That is, if F is a family of graphs such that H ∈ F and G ≤ H implies
G ∈ F, then there exists a finite set of graphs {G1 , . . . , Gm } such that G ∈ F if, and
only if, Gi 6≤ G for all i ≤ m. Together with the observation that for a fixed graph
G, determining if G is a minor of a given graph can be decided in cubic time, this we
obtain the following important algorithmic consequence.
Corollary 8.43 ([RS04]). Let F be a minor-closed family of graphs. The problem of
deciding if G ∈ F can be computed in cubic time.
Thus it is an interesting problem to see if we can generalize the Graph Minor Theorem to directed graphs. Unfortunately, for most of the minor relations we have defined,
this is not the case.
Theorem 8.44. E and S are not well-quasi orders.
Proof. Consider the sequence of bidirected cycles C3 , C4 , C5 , . . . pictured in Figure 8.3.
Using the same argument as in the proof of Theorem 8.32, Example (V), it is easy to
see that Ci E
6 Cj for i < j. Thus E is not a well-quasi order.
Now consider the sequence of graphs C4⊕ , C6⊕ , . . . pictured in Figure 8.4. It is
easy to see that for all even i ≥ 4, an edge in Ci⊕ can neither be butterfly contracted
nor strong contracted, and the deletion of any edge results in a graph with an acyclic
underlying graph. Thus for all i < j, Ci⊕ 6S Cj⊕ , and so S is not a well-quasi order.
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CHAPTER 8. HAVENS, BRAMBLES AND MINORS
170
•O o
•
C4⊕
•
/•
/
•
•X1
C6⊕
11
1
/
•
•X1
11
1
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•
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•


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??
•
C8⊕
• ??
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•
•
?


Figure 8.4: An infinite anti-chain for the S relation
···
Chapter 9
Conclusion and Future work
In this dissertation we examined the role of infinite games on finite graphs in two
aspects of complexity: computational complexity and structural complexity. The research resolved some unanswered questions in the literature and opened up some interesting avenues for further research. We conclude this dissertation by recalling the
major results established, and discussing possible areas for future study.
9.1 Summary of results
In Chapter 1 we stated the two main goals of this dissertation: to investigate the computational complexity of infinite games on finite graphs, and to use infinite games to
define an algorithmically useful notion of structural complexity for directed graphs.
The first of these goals was predominantly addressed in Chapters 2, 3, 6 and 7, while
the second was catered for in Chapters 4 to 8. We now summarize the contribution
each chapter made to each goal.
Complexity of Infinite Games
In Chapter 2 we considered the general class of infinite games on finite graphs. We
introduced a generalization of bisimulation called game simulation which enables us
to translate strategies from one game to another. We then introduced the notion of a
condition type, which gives us a general framework for comparing many types of games
which occur in the literature, for example Muller games [Mul63], Rabin games [Rab72],
Streett games [Str82] and parity games [Mos91, EJ91]. The notion of translatability
between condition types lets us compare the computational complexity of two games
via the expressibility and succinctness of their winning conditions. We considered the
computational complexity of deciding the winner in Muller games. We provided polynomial time algorithms for explicitly presented Muller games under various restrictions
on the family of sets which specified the winning condition, namely simple games, and
games where the condition is an anti-chain. We showed that deciding the winner of
win-set games was P SPACE -complete. Following our work on translatability, it follows
171
172
CHAPTER 9. CONCLUSION AND FUTURE WORK
that the problems of deciding the winner of Muller games where the winning condition
is specified as a Muller, Zielonka DAG, Emerson-Lei, or a circuit condition are all also
P SPACE -complete, thus closing one of the open problems relating to the complexity of
Muller games that we discussed in Chapter 1. We showed that the completeness results
carries over to arenas of bounded tree-width for games specified by a Muller condition.
We also gave examples of union-closed and upward-closed games for which deciding
the winner is co-NP-complete. We ended the chapter by showing how the lower bounds
for deciding win-set games can be used to establish that the non-emptiness and modelchecking problems for Muller automata are also P SPACE-complete, thereby resolving
an open question in the field of automata theory.
Our foray into the sticky world of parity games began in Chapter 3, where we analysed one of the best performing algorithms for deciding parity games in an effort to
establish tighter bounds on the running time. We interpreted the algorithm from a combinatorial perspective, in particular as a method for finding a global sink on an acyclic
unique sink oriented hypercube. Using techniques from combinatorics, we improved
the upper bound for the running time. We also provided an example which shows that
the hypercube orientations resulting from parity games are not pseudomodular.
In Chapters 6 and 7, we demonstrated how the structure of the arena affects the
complexity of deciding the winner of parity games. We used DAG-decompositions in
Chapter 6 and Kelly-decompositions in Chapter 7 to produce two dynamic programming style algorithms for solving parity games. The upshot of such algorithms is that
on a class of arenas of bounded DAG-width or bounded Kelly-width, there is a polynomial time algorithm for deciding the winner of a parity game. As DAG-width and
Kelly-width encompass other graph parameters such as tree-width, this gives us the
largest class of graphs so far known on which parity games can be solved in polynomial time.
Complexity by Infinite Games
In Chapter 4 we discussed the properties that a good measure of digraph structural
complexity should have. We cited tree-width as an example to aspire towards, and
discussed why tree-width is not suitable as a measure for directed graphs. We also
discussed why the established notion of directed tree-width from [JRST01] is also not
entirely suitable.
In Chapter 5 we introduced a framework for defining reasonable structural complexity measures via graph searching games, a form of the infinite games we have
been considering. We showed how these games encompass many similar games in the
literature, including those that can be used to characterize tree-width.
In Chapter 6 we used the work from Chapter 5 to define an extension of tree-width
to directed graphs, DAG-width. Unlike directed tree-width and Kelly-width, the definition of a DAG-decomposition closely resembles tree decompositions. After showing
that cop-monotonicity and robber-monotonicity coincide in this game, we showed that
DAG-width is equivalent to the number of cops required to capture a visible robber with
a monotone strategy, thereby demonstrating that it is a reasonable measure of structural
complexity for directed graphs. We also showed that DAG-width defines an algorithmically useful complexity measure by showing that a number of problems, including
9.2. FUTURE WORK
173
deciding the winner of a parity game, can be solved in polynomial time on graphs of
bounded DAG-width. We concluded the chapter by demonstrating that DAG-width
is markedly different from three other measures defined in the literature: tree-width,
directed tree-width and directed path-width.
In Chapter 7 we considered the generalization to directed graphs of three characterizations of tree-width: partial k-trees, elimination orderings and the cops and inert
robber graph searching game. This results in partial k-DAGs, directed elimination
orderings, and the cops and inert robber game for directed graphs. We showed that
the graph parameters defined by these three generalizations were all equivalent, and
these, in turn, were equivalent to the width of a decomposition we introduced called a
Kelly-decomposition. As with DAG-width, we demonstrated the algorithmic potential
of Kelly-width by exhibiting polynomial time algorithms for a number of problems,
including deciding the winner of a parity game, on graphs of bounded Kelly-width. We
concluded the chapter by showing that, as with DAG-width, Kelly-width is quite different from tree-width, directed tree-width and directed path-width. However, its relation
to DAG-width is somewhat more complex. We showed that, in the graph searching
games which characterize DAG-width and Kelly-width, a monotone winning strategy
for the cops in one game implies a winning strategy in the other (with possibly twice
as many cops). Without a result in either game relating the number of cops required
for a monotone strategy to the number of cops with a winning strategy, we are unable
to compare DAG-width and Kelly-width directly. However, we do show that there are
graphs on which DAG-width and Kelly-width differ (by an arbitrary amount).
Finally, in Chapter 8 we presented preliminary results towards a directed graph
structure theory, based on the notions of structural complexity we have developed. We
introduced generalizations of havens and brambles which appear to correspond with
DAG-width and Kelly-width. The brambles for DAG-width are dual to the brambles for
Kelly-width, suggesting that DAG-width and Kelly-width are very closely connected.
We also considered the problem of extending the minor relation to directed graphs.
We introduced a number of distinct relations ranging from the subgraph relation to the
minor relation on the underlying undirected graphs. We showed that these relations
do not enjoy the algorithmic properties of the minor relation, as deciding if a fixed
subgraph is a minor of a given graph is, in general, NP-complete for most of the minor
relations we considered. We concluded the chapter by showing that all except two
of the minor relations we introduced contain infinite anti-chains. This implies that to
consider a generalization of the Graph Minor Theorem using the minor relations we
defined, we need to use either the anti-parallel D-minor or the strong D-minor relation.
9.2 Future work
The work we have presented in this dissertation raises a number of interesting questions
and directions for further research. We now discuss some of these, roughly in the order
they arose during the dissertation.
The exact complexity for deciding Muller games when the winning condition is
explicitly presented remains open, as does the question for union-closed games with
an explicitly presented winning condition. We saw in Theorem 2.62 that if the winning
174
CHAPTER 9. CONCLUSION AND FUTURE WORK
condition is an anti-chain then the game can be solved efficiently. Thus it is possible
that the complexity of the former problem can be derived from the complexity of the
latter. This would also be an interesting question to investigate.
The exact complexity for deciding parity games also remains an interesting open
problem. Characterizing the acyclic unique sink orientations that arise from valuations
in parity games could either establish a polynomial time algorithm for parity games, or
give a super-polynomial lower bound for the strategy improvement algorithm.
Monotonicity questions frequently arise in the study of graph searching games. An
interesting line of research would be to characterize the properties of graph searching
games necessary for monotonicity to be sufficient. For example, extending the work
of Fomin and Thilikos [FT03]. On a more specific level, for the cops and visible robber game on directed graphs an important open problem is finding a relation between
the number of cops required for a monotone winning strategy and the number of cops
required for a winning strategy which is not necessarily monotone. Such a correspondence allows us to compare DAG-width with other parameters we have considered
such as D-havens and Kelly-width. Similarly, finding a relation between the number of
cops required for a robber-monotone winning strategy and the number of cops required
for a not necessarily monotone winning strategy in the inert robber game allows us to
compare Kelly-width to other measures.
Two important questions regarding the complexity of DAG-width and Kelly-width
still remain open. First is the question of whether deciding if a digraph has DAG-width
at most a given integer is in NP. Second is the question of whether, for a fixed k if
deciding whether a digraph has Kelly-width at most k is decidable in polynomial time.
An improved bound from O(nk ) on the size of a DAG-decomposition of a graph would
benefit the first question.
Finally, the preliminary work on a structure theory based on directed connectivity
raises a number of interesting questions. For example, determining the precise relationship between DAG-width, Kelly-width, and initial and terminal brambles; characterizing the pattern graphs G for which (G, )-M INOR is solvable in polynomial time;
determining if any of the introduced minor relations is a well-quasi order; and characterizing classes of graphs via forbidden minors.
9.3 Conclusion
In conclusion, this dissertation has made a significant contribution towards the analysis of the complexity of infinite games and to the development of a notion of structural
complexity for directed graphs, and opened up exciting possibilities for future research.
We resolved the open questions regarding the exact complexity of deciding Muller
games and Muller automata non-emptiness and model-checking, and we made substantial progress towards answering the question for parity games. We introduced two
similar measures of structural complexity for directed graphs which appear to measure
the directed connectivity of a digraph, a metric which lies between weak connectivity
and strong connectivity and is distinct from both. We demonstrated their algorithmic
benefits by providing efficient algorithms for problems not known to be decidable in
polynomial time.
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