nauty User's Guide (Version 2.4) - Computer Science

nauty User's Guide (Version 2.4) - Computer Science

nauty User’s Guide (Version 2.4)

Brendan D. McKay

Department of Computer Science

Australian National University

Canberra ACT 0200, Australia [email protected]

November 4, 2009

Contents

0 . How to use this Guide.

1 . Introduction.

2 . Outline of the algorithm.

3 . Data structures.

4 . Size limits.

5 . Description of the procedure parameters.

6 . Interpretation of the output.

7 . Examples.

8 . User-defined procedures.

9 . Vertex-invariants.

10 . Writing programs which call nauty.

11 . Programming with sparse representation.

12 . Variations.

13 . Installing nauty and dreadnaut.

14 . Efficiency.

15 . The dreadnaut program.

16 . gtools.

17 . Recent changes.

18 . Sample programs which call nauty.

19 . Graph formats used by gtools programs.

20 . Help texts for gtools programs.

References.

Research supported by the Australian Research Council.

1

0 How to use this Guide.

The dreadnaut program provides sufficient functionality that most simple applications can be managed without the need to write any programs. Section

15

is intended to be a fairly self-contained introduction to that level of use. You should start reading there; it will direct you to any necessary information which appears elsewhere.

If you wish to write C programs which call nauty, you don’t have much choice but to read this Guide from start to finish. However, it isn’t really as hard as it sounds; see the examples in Section

18

for a constructive proof.

The current version of nauty is available at http://cs.anu.edu.au/

∼ bdm/nauty

.

There is a mailing list you can subscribe to if you want to discuss nauty and receive upgrade notices: http://dcsmail.anu.edu.au/cgi-bin/mailman/listinfo/nauty-list

.

1 Introduction.

nauty (no automorphisms, yes?) is a set of procedures for determining the automorphism group of a vertex-coloured graph. It provides this information in the form of a set of generators, the size of the group, and the orbits of the group. It is also able to produce a canonically-labelled isomorph of the graph, to assist in isomorphism testing.

The mathematical basis for the algorithm is described in [ 9 ]; only a broad outline is given

here. Note, however, that a great number of improvements have been made since the

implementation described in [ 9 ].

Useful ideas received from Greg Butler, Aaron Grosky, Andrew Kirk, Bill Kocay, Rudi

Mathon, Kevin Malysiak, Mark Henderson, Gordon Royle, Carsten Saager, Neil Sloane,

Paulette Lieby and several others, are gratefully acknowledged. I’d also like to thank

Paul Darga, Mark Liffiton, Karem Sakallah and Igor Markov for prompting me to hurry up with the sparse implementation, and perhaps for some ideas I found in their program

saucy [

2 ].

The author would appreciate receiving any comments about the program and/or this

Guide, especially about apparent bugs.

nauty is written in a highly portable subset of the language C. Modern C compilers for most types of computer should be able to handle nauty without difficulty.

2 The Algorithm.

Throughout this document, a graph is a simple graph with n vertices labelled 0, 1, . . . , n−1.

Digraphs, and graphs with loops, can also be handled correctly (see Section

5 ), but we

will not mention them much. The vertex set of a graph G is denoted by V = V (G).

The terms colouring and partition will be used interchangeably to denote a partition of V into disjoint non-empty colour classes or cells. The order of the cells is significant,

2

but the order of the vertices within each cell is not. If π is finer than π

2

, and π

2 is coarser than π

1

1 and π

2

, if every cell of π

1 are partitions, then π

1 is a subset of some cell of

π

2

. (Note that partitions are both finer and coarser than themselves.) A singleton cell is a cell with cardinality one, while a discrete partition is one with only singleton cells.

Let G be a graph, γ a permutation of V , v ∈ V , W ⊆ V , and π = (V

0

, V

1

, . . . , V k

) a partition of V . Then v

γ is the image of v under γ, W

γ

= {w

γ

| w ∈ W }, G

γ is the graph in which vertices x

γ is the partition (V

0

γ and y

, V

1

γ

γ are adjacent if and only if x and y are adjacent in G, and π

, . . . , V

γ k

).

γ

The automorphism group of a coloured graph (G, π) is the set of all permutations γ such that G

γ

= G and π

γ

= π. Since the order of cells in partitions is significant, the last condition means that γ fixes each cell of π setwise (i.e., γ is colour preserving). In the majority of applications, π has only one cell V , so we get the usual automorphism group.

If π = (V

0

, V

1

, . . . , V k

) is a partition of {0, 1, . . . , n−1}, then c(π) is the partition

({0, 1, . . . , |V

0

|−1}, {|V

0

|, . . . , |V

0

| + |V

1

| − 1}, . . . , {n − |V k

|, . . . , n−1}). Thus, c(π) has the same cell sizes as π, in the same order, but is otherwise independent of π.

A canonical labelling map is a function C such that, for any graph G, partition π of V , and permutation γ of V , we have

(a) C(G, π) = G

δ for some permutation δ such that π

δ

(b) C(G

γ

, π

γ

) = C(G, π).

= c(π), and

Informally, C relabels the vertices of G in order of colour, ignoring the original vertex labels. The usefulness of a canonical labelling map is as follows.

Theorem 1. Suppose the graphs G tices of each colour. Then C(G permutation γ. (Here, π

1

1 and π

, π

2

1

1 and G

) = C(G

2

2 are coloured using the same number of ver-

, π

2

) iff G

γ

1

= G

2 for some colour-preserving are the colourings, with the colours in the same order in each.)

Let G be a graph and π a partition of V with cells V

0

, V

1

, . . . , V k

. Then π is equitable

(with respect to G) if there are numbers d ij such that each vertex in V i is adjacent to precisely d ij vertices in V j

, for 0 ≤ i, j ≤ k. Up to the order of the cells, there is a unique coarsest equitable partition which is finer than any given partition.

A refinement function is a function R such that, for any graph G, partition π of V , and permutation γ of V , we have

(a) R(G, π) is a partition of V which, up to the order of the cells, is the coarsest equitable partition finer than π, and

(b) R(G

γ

, π

γ

) = R(G, π)

γ

.

The algorithm used by nauty is a backtrack program which can be described in terms of the usual associated search tree. We will refer to the nodes of the tree to avoid confusion with the vertices of G. The root of the tree is associated with the initial colouring π of G and the equitable partition π

0

= R(G, π). If π

0 is discrete, the automorphism group is trivial and we can obtain C(G, π) by labelling the vertices of G in the order that they appear in π

0

. Suppose more generally that the equitable partition π

0 is associated with some node ν of the tree. If π

0 is discrete, then ν has no children. If π

0 is not discrete, let C be a non-singleton cell of it. This is called the target cell for this node (chosen by

3

nauty according to some rule). For each vertex v ∈ C we have a child of ν associated with the partition got from π

0 by replacing the cell C by the pair of cells {v} and C − {v}, in that order, and the equitable partition obtained by applying R to it. The children of ν are generated in ascending order of the labels on the vertices of C.

Any node of the tree for which the equitable partition is discrete corresponds to a labelling of G, as described above. Automorphisms of the graph are found by noticing that two such labellings give the same labelled graph. The canonical labelling map corresponds to one of these labellings, chosen according to a complicated scheme for which you will

have to consult [ 9 ] or the source code.

Except in particularly simple cases, only some of the tree is actually generated. The other parts of the tree are either shown to be equivalent to parts already generated, or

shown to be uninteresting. Again, see [ 9 ] for details.

In Figure One, we show an example of the part of the tree which is actually generated. The nodes are represented by their equitable partitions, assuming that the original colouring only used one colour. The target cells are underlined and the numbers on the tree edges give the elements of the target cells which are being fixed. In this example, all the leaves are equivalent and correspond to the automorphisms (1), (1 2)(4 5), and

(0 1)(3 4), respectively.

4

1

3

0 2

5

[3 4 5 | 0 1 2]

3

[3 | 4 5 | 1 2 | 0]

4

[4 | 3 5 | 0 2 | 1]

4

3

5

[3 | 4 | 5 | 2 | 1 | 0] [3 | 5 | 4 | 1 | 2 | 0] [4 | 3 | 5 | 2 | 0 | 1]

Figure 1: Example of search tree generated by nauty

3 Data Structures.

A setword is an unsigned integer type of either 16, 32 or 64 bits, depending on the compile-time parameter WORDSIZE. (By default, WORDSIZE is 32 unless the size of type long int is greater than 32, in which case WORDSIZE is 64.)

4

A set (by which we always mean a subset of V = {0, 1, . . . , n−1}) is represented by an array of m setwords, where m is some number such that WORDSIZE × m ≥ n. The bits of a set are numbered 0, 1, . . . , n−1 left to right (within each setword: high order to low order). Bits which don’t get numbers are called “unnumbered” and are assumed permanently zero. A set represents the subset { i | bit i is 1 }.

There are two ways of representing a graph, which we will call the packed form and the sparse form.

A graph represented in packed form uses the type graph. It is stored as an array of n sets (so it has mn setwords altogether). The i-th set gives the vertices to which vertex i is adjacent, for 0 ≤ i < n.

A graph represented in sparse form uses the type sparsegraph. It is stored as a structure with the following fields: int nv : the number of vertices int nde: the number of directed edges (loops count as 1) int ∗v : pointer to an array of length at least nv int ∗d : pointer to an array of length at least nv int ∗e: pointer to an array of length at least nde

SG WEIGHT ∗w : not implemented in this version, should be NULL size t vlen, dlen, elen, wlen: the actual lengths of the arrays v, d, e and w. The unit is the element type of the array in each case (so vlen is the number of ints in the array v, etc.)

For each vertex i = 0 . . . n−1, d[i] is the degree (out-degree for a digraph) of that vertex. v[i] is an index into the array e such that e[v[i]], e[v[i] + 1], . . . , e[v[i] + d[i] − 1] are the vertices to which vertex i is joined. It is not necessary that this list of neighbours be sorted. These neighbour lists can be present in the array e in any order and may have gaps between them, but cannot overlap. If d[i] = 0 for some i, v[i] is not used.

1

0

2

0110 · · · 0

1100 · · · 0

1000 · · · 0 nv 3 nde 5 d

% v

−→ e dlen 5

& vlen 3 elen 7

2 2 1

0 3 5

1 2 0 1 0

Figure 2: Packed and sparse data structures for graphs.

In Figure

2 , the graph on the left is represented in packed form by the array in the

centre (we show three words of type setword). On the right is a possible sparse form for the same graph.

5

The C types setword, set and graph are actually the same, so a graph in packed form is really represented by a 1-dimensional array of length mn, not by an array of arrays.

A permutation of V is represented by an array of n integers, the i-th entry giving the image of i under the permutation. The type of the entries, permutation is the same as int but has its own type name for historical reasons.

The type boolean is a synonym for int, but the different name is intended to encourage you to restrict the values to either TRUE or FALSE (which are defined as 1 and 0, respectively).

The structured types optionblk and statsblk are described below. All these types are defined in the file nauty.h.

Note that types like set actually refer to the elements of the arrays (in this case setword) rather than the arrays themselves. This is done because the lengths of the arrays are not known in advance. We use set rather than setword purely for self-documentation purposes.

4 Size limits.

There are several ways to compile nauty, leading to differences in types and the size of graph that can be processed. These are selected by preprocessor variables.

(1) If type int has less than 32 bits (rare these days), there is an absolute limit of

2

15

− 3 = 32765.

(2) If type int has at least 32 bits, there is an absolute limit of 2

30 the order of a graph.

= 1073741824 on

In addition, there is a choice between static and dynamic memory allocation for the larger data objects. This is selected by the value of the preprocessor variable MAXN.

(a) If MAXN is defined as 0, the limit on the order of a graph is given in (1)–(2) above and objects are dynamically allocated. Of course, if you don’t have enough memory, dynamic allocation may fail. This is the default.

(b) If MAXN is defined as a positive integer, that is the limit on the order of a graph. It can’t be greater than the absolute limit given in (1)–(2) above. In this case objects are statically allocated, so space is wasted if MAXN is much larger than what is actually used.

A special case of option (b) is 0 < MAXN ≤ WORDSIZE, which implies that a set consists of a single setword. Some of the critical routines in nauty have special code to optimize performance in that case. The recommended way to compile for this case is to define MAXN to be the name WORDSIZE.

6

5 Parameters.

A call to nauty has the form nauty (g, lab, ptn, active, orbits, options, stats, workspace, worksize, m, n, canong) where the parameters have meanings as defined below.

graph or sparsegraph ∗g :

The input graph. Read-only.

int ∗lab,∗ptn:

Two arrays of n entries. Their use depends on the values of several options. If options.defaultptn = TRUE, the input values are ignored; otherwise, they define the initial colouring of the graph (see below). If options.getcanon =

TRUE, the value of lab on return is the canonical labelling of the graph. Precisely, it lists the vertices of g in the order in which they need to be relabelled to give canong. Irrespective of options.getcanon, neither lab nor ptn is changed by enough to change the colouring. (Recall that the order of the vertices within the cells is irrelevant.) Read-Write.

set ∗active:

An array of m setwords specifying the colours which are initially active.

A brief outline of what this means is given below. This argument is rarely used; nauty will always work correctly if given the nil pointer NULL. Read-only.

int ∗orbits:

An array of n entries to hold the orbits of the automorphism group. When nauty returns, orbits[i] is the number of the least-numbered vertex in the same orbit as i, for 0 ≤ i ≤ n−1. Write-only.

optionblk ∗options:

A structure giving a list of options to the procedure. See below for their meanings. Read-only.

statsblk ∗stats:

A structure used by nauty to provide some statistics about what it did. See below for their meanings. Write-only.

setword ∗workspace, worksize:

The address and length of an integer array used by nauty for working storage. There is no minimum requirement for correct operation, but the efficiency may suffer if not much is provided. A value of worksize ≥ 50m is recommended. Write-only and read-only, respectively.

int m, n:

The number of setwords in sets and the number of vertices, respectively.

It must be the case that 1 ≤ n ≤ m × WORDSIZE. If nauty is compiled with

MAXN > 0, it must also be the case that n ≤ MAXN and m ≤ MAXM, where

MAXM = dMAXN/WORDSIZEe. Read-only.

graph or sparsegraph ∗canong

The canonically labelled isomorph of g produced by nauty. This argument is ignored if options.getcanon = FALSE, in which case the nil pointer NULL can be given as the actual parameter. Write-only. The type must be the same as that of parameter g.

The C type of the parameters g and canong is graph∗. If another type of pointer is passed (for example sparsegraph∗), it should be cast to type graph∗.

The initial colouring of the graph is determined by the values of the arrays lab, ptn and the flag options.defaultptn. If options.defaultptn = TRUE, the contents of lab and ptn are set by nauty so that every vertex has the same colour. If not, they are assumed

7

to have been set by the user. In this case, lab should contain a list of all the vertices in some order such that vertices with the same colour are contiguous. The ends of the colour-classes are indicated by zeros in ptn. In super-precise terms, each cell has the form {lab[i], lab[i+1], . . . , lab[j]} where [i, j] is a maximal subinterval of [0, n−1] such that ptn[k] > 0 for i ≤ k < j and ptn[j] = 0. (In the terminology defined in Section

8 , this is

the “partition at level 0”.) The order of the vertices within each cell has no effect on the behaviour of nauty. An example is given in Section

7 .

The concept of active cells is used by the procedure which implements the partition

refinement function R defined above. The details are given in [ 9 ], where the active cells

are in a sequence called α. In this implementation, a set rather than a sequence is used.

If options.defaultptn = TRUE, or active = NULL, every colour is active. This will always work, and so is recommended if you don’t want to be a smart-arse. If options.defaultptn

= FALSE and active 6= NULL, the elements of active indicate the indices (0 .. n−1) where

the active cells start in lab and ptn (see above). Theorem 2.7 of [ 9 ] gives some sufficient

conditions for active to be valid. If these conditions are not met, anything might happen.

The most common places where this feature may save a little time are:

(a) If the initial colouring is known to be already equitable, active can be the empty set.

(Don’t confuse this with NULL, which causes nauty to set the active set to include every cell.)

(b) If the graph is regular and the colouring has exactly two cells, active can indicate just one of them (the smallest for best efficiency).

If nauty is used to test two graphs for isomorphism, it is essential that exactly the same value of active be used for each of them.

The various fields of the structure options fine-tune the behaviour of nauty. The recommended way to assign values to these options is to start with the declaration

DEFAULTOPTIONS GRAPH(options); - for packed graphs

DEFAULTOPTIONS SPARSEGRAPH(options); - for sparse graphs

These define the static variable options of type optionblk, initialized to sensible values for most circumstances. Changes in those values can then be made using assignment statements, for example options.linelength = 100;

This practice will protect your code from breaking if additional fields are added to options in future editions of nauty, which is quite likely. (You should just need to recompile.)

All of these fields are read-only.

boolean getcanon:

If this is TRUE, the canonically labelled isomorph canong is produced, and lab is set to indicate the canonical label, as described above. Otherwise, only the automorphism group is determined. Sometimes, different generators of the automorphism group are found if this option is selected; of course, the group they generate is the same. Default FALSE.

boolean digraph:

This must be TRUE if the graph has any directed edges or loops. It has the effect of turning off some heuristics which are only valid for simple graphs.

If no directed edges or loops are present, selecting this option is legal but may

8

degrade the performance slightly. Default FALSE.

boolean writeautoms:

If this is TRUE, generators of the automorphism group will be written to the file outfile (see below). The format will depend on the settings of options cartesian and linelength (see below, again). More details on what is written can be found in Section

6 . Default FALSE (changed with version 2.1).

boolean writemarkers:

If this is TRUE, extra data about the automorphism group generators will be written to the file outfile (see below). An explanation of what these data are can be found in Section

6 . Default FALSE (changed with version 2.1).

boolean defaultptn:

This has been fully explained above. Default TRUE.

boolean cartesian:

If writeautoms = TRUE, the value of this option effects the format in which automorphisms are written. If cartesian = FALSE, the output is the usual cyclic representation of γ, for example “(2 5 6)(3 4)”. If cartesian = TRUE, the output for an automorphism γ is the sequence of numbers “1

γ

2

γ

. . . (n−1)

γ

”, for example “1 5 4 3 6 2”. Default FALSE.

int linelength:

The value of this variable specifies the maximum number of characters per line (excluding end-of-line characters) which may be written to the file outfile

(see below). Actually, it is ignored for the output selected by the option writemarkers, but that never has more than about 65 characters per line anyway. A value of 0 indicates no limit. Default CONSOLWIDTH, which can be defined when compiling but is set to 78 otherwise.

FILE ∗outfile:

This is the file to which the output selected by the options writeautoms and writemarkers is sent. It must be already open and writable. The nil pointer

NULL is equivalent to stdout (the standard output). Default NULL.

void (∗userrefproc)():

This is a pointer to a user-defined procedure which is to be called in place of the default refinement procedure. Section

8

has details. If the value is

NULL, the default refinement procedure is used. Default NULL.

void (∗userautomproc)():

This is a pointer to a user-defined procedure which is to be called for each generator. Section

8

has details. No calls will be made if the value is NULL. Default NULL.

void (∗userlevelproc)():

This is a pointer to a user-defined procedure which is to be called for each node in the leftmost path downwards from the root, in bottom to top order. Section

8

has details. No calls will be made if the value is NULL. Default

NULL.

void (∗usernodeproc)():

This is a pointer to a user-defined procedure which is to be called for each node of the tree. Section

8

has details. No calls will be made if the value is NULL. Default NULL.

void (∗invarproc)():

This is a pointer to a vertex-invariant procedure. See Section

9

for a discussion of vertex-invariants. No calls will be made if the value is NULL.

Default NULL.

int tc level :

Two rules are available to choose target cells. On levels up to level tc level , inclusive, an expensive but (empirically) highly effective rule is used. (The root of

9

the search tree is at level one.) At deeper levels, a cheaper rule is used. For difficult graphs, a large value is recommended. For easier graphs, use 0. Default 100.

int mininvarlevel :

The absolute value gives the minimum level at which invarproc will be applied. (The root of the search tree is at level one.) If options.getcanon =

FALSE, a negative value indicates that the minimum level will be automatically set by nauty to the least level in the left-most path in the search tree where invarproc is applied and refines the partition. If options.getcanon = TRUE, the sign is ignored.

A value of 0 indicates no minimum level. Default 0.

int maxinvarlevel :

The absolute value gives the maximum level at which invarproc will be applied. (The root of the search tree is at level one.) If options.getcanon =

FALSE, a negative value indicates that the maximum level will be automatically set by nauty to the least level in the left-most path in the search tree where invarproc is applied and refines the partition. If options.getcanon = TRUE, the sign is ignored.

A value of 0 effectively disables invarproc. Default 1 (changed with version 2.1).

int invararg :

This level is passed by nauty to the vertex-invariant procedure invarproc, which might use it for any purpose it pleases. Default 0.

dispatchvec ∗dispatch:

This is a vector of procedure pointers used to apply nauty to objects other than graphs. Version 2.4 only has full support for graphs in packed and sparse form. NULL is not permitted as a value (unlike in versions before 2.3).

The macro DEFAULTOPTIONS GRAPH sets dispatch to point to the vector dispatch graph defined in nauty.c. The macro DEFAULTOPTIONS SPARSEGRAPH sets dispatch to point to the vector dispatch sparse defined in nausparse.c.

void ∗extra options:

This currently does not have a default use, and should be NULL.

Some of the fields in the options argument may change the canonical labelling produced by nauty. These are fields digraph, defaultptn, tc level, userrefproc, invarproc, mininvarlevel, maxinvarlevel, invararg and dispatch. The canonical labelling also depends on whether the graph is in packed or sparse form. If nauty is used to test two graphs for isomorphism, it is important that the same values of these options be used for both graphs.

The various fields of the structure stats are set by nauty. Their meanings are as follows: double grpsize1 , int grpsize2 :

Within rounding error, the order of the automorphism group is equal to grpsize1 × 10grpsize2 . If the exact size of a very large group is needed, it can be calculated from the output selected by the writemarkers option, or you can compute it with your own multiprecision arithmetic using the userlevelproc feature. See Section

6 .

int numorbits:

The number of orbits of the automorphism group.

int numgenerators:

The number of generators found.

int errstatus:

If this is nonzero, an error was detected by nauty. Possible values are:

• MTOOBIG: m is too big; i.e., the maximum is 1073741826/WORDSIZE+1 if

MAXN=0 and int has at least 32 bits, 32765/WORDSIZE+1 if MAXN=0 and

10

int has at least 32 bits, and dMAXN/WORDSIZEe otherwise.

• NTOOBIG: n is too big. Either n > WORDSIZE × m or n exceeds its absolute limit as in Section

4 .

• CANONGNIL: canong = NULL, but options.getcanon = TRUE.

nauty also writes a message to stderr in these cases, so there is no real need to test this parameter in most applications.

unsigned long numnodes:

The total number of tree nodes generated.

unsigned long numbadleaves:

The number of leaves of the tree which were generated but were useless in the sense that no automorphism was thereby discovered and the current-best-guess at the canonical labelling was not updated.

int maxlevel :

The maximum level of any generated tree node. The root of the tree is on level one.

unsigned long tctotal :

The total size of all the target cells in the search tree. The difference between this value and numnodes provides an estimate of the efficiency of nauty’s search-tree pruning.

unsigned long canupdates:

The number of times the program’s idea of the “best candidate for canonical label” was updated, including the original one.

unsigned long invapplics:

The number of nodes at which the vertex-invariant was applied.

unsigned long invsuccesses:

The number of nodes at which the vertex-invariant succeeded in refining the partition more than the refinement procedure did.

int invarsuclevel :

The least level of the nodes in the tree at which the vertex-invariant succeeded in refining the partition more than the refinement procedure did. The value is zero if the vertex-invariant was never successful.

The values corresponding to node counts might overflow during a long computation, but this is not a serious problem as they have no effect on the computation at all.

In addition to their parameters, the output routines of nauty respect the value of the global int variable labelorg. If the value of labelorg is k, the output routines pretend that the vertices of the graph are numbered k, k+1, . . . , n+k−1, even though they are internally numbered 0, 1, . . . , n−1. By default, k = 0. Only non-negative values are supported.

6 Output.

If options.writeautoms = TRUE or options.writemarkers = TRUE, information concerning the automorphism group is written to the file options.outfile.

Let Γ be the automorphism group, and let Γ v

1

,v

2

,...,v i denote the point-wise stabiliser in Γ of v

1

, v

2

, . . . , v i

. The output has the following general form:

11

(k)

γ

1

(k)

γ

2

.

.

.

(k)

γ t k level k: c k cells; r k orbits; v k fixed; index i k

/j k

(k−1)

γ

1

(k−1)

γ

2

.

.

.

(k−1)

γ t k−1 level k−1: c k−1

.

.

.

cells; r k−1 orbits; v k−1 fixed; index i k−1

/j k−1 level 2: c

2

(1)

γ

1

(1)

γ

2

.

.

.

(1)

γ t

1 level 1: c

1 cells; r cells; r

2

1 orbits; v orbits; v

2

1 fixed; index i fixed; index i

2

1

/j

/j

2

1

Here, v

1

, v

2

, . . . , v k is a sequence of vertices such that Γ v

1

,v

2

,...,v k automorphisms. For 1 ≤ l ≤ k, the following are true.

(j)

(a) Γ v

1

,v

2

,...,v l−1 is generated by the automorphisms γ i

(b) Γ v

1

,v

2

,...,v l−1 has r l orbits and order i

1 i

2

· · · i l

.

(c) c l is trivial. The γ for l ≤ j ≤ k and 1 ≤ i ≤ t i

(j) j

.

are is the number of cells in the equitable partition at the ancestor at level l of the first leaf of the tree, j l is the number of vertices in the target cell of the same node, v l is the first vertex in that cell, and i l is the number of vertices of that cell which are equivalent to v l

.

P k

(d) i=l t i

≤ n − r l

. This follows from the fact that the number of orbits of the group generated by all the automorphisms found to up to any moment decreases as each new automorphism is found. In particular, this means that the total number of generators found is at most n−1. Usually, it is much less.

The markers “level...” are only written if options.writemarkers = TRUE. In the common circumstance that c j l

= i l

. Note that i l l

= r l

, “c l cells;” is omitted. Similarly, “/j l

” is omitted if

= 1 is possible for more difficult graphs. Further information about

the generators can be found in Theorem 2.34 of [ 9 ].

12

7 Examples.

All of the following examples were run without the use of a vertex-invariant.

Example 1:

0 1

3

2

g =

4

7

6

5

options[getcanon = FALSE, digraph = FALSE, writeautoms = TRUE, writemarkers =

TRUE, defaultptn = TRUE, cartesian = FALSE, tc level = 0].

output:

(2 5)(3 4) level 3: 6 orbits; 3 fixed; index 2

(1 3)(5 7) level 2: 4 orbits; 1 fixed; index 3

(0 1)(2 3)(4 5)(6 7) level 1: 1 orbit; 0 fixed; index 8 orbits = (0,0,0,0,0,0,0,0), stats[grpsize1 = 48.0, grpsize2 = 0, numorbits = 1, numgenerators = 3, numnodes = 10, numbadleaves = 0, maxlevel = 4].

Explanation of output: Let γ

1

, γ

2 and γ

3 be the three automorphisms found, in the order written. Let Γ be the automorphism group. Then

Γ

0,1,3

= {(1)}

Γ

0,1

= hγ

1 i with 6 orbits and order 2

Γ

0

= hγ

1

, γ

2 i with 4 orbits and order 2 × 3 = 6

Γ = hγ

1

, γ

2

, γ

3 i with 1 orbit and order 6 × 8 = 48.

Example 2:

0 1

3

2

g =

4

7

6

5

lab = (2,0,1,3,4,5,6,7), ptn = (0,1,1,1,1,1,1,0), active = NULL, options[getcanon = FALSE, digraph = FALSE, writeautoms = TRUE, writemarkers =

TRUE, defaultptn = FALSE, cartesian = TRUE, tc level = 0].

output:

5 1 2 6 4 0 3 7 level 2: 6 orbits; 3 fixed; index 2

0 3 2 1 4 7 6 5 level 1: 4 orbits; 1 fixed; index 3

13

orbits = (0,1,2,1,4,0,1,0), stats[grpsize1 = 6.0, grpsize2 = 0, numorbits = 4, numgenerators = 2, numnodes = 6, numbadleaves = 0, maxlevel = 3].

In this example we have set lab, ptn and options.defaultptn so that vertex 2 is fixed.

The automorphisms were written in the “cartesian” representation, which would probably only be useful if they were going to be fed to another program. The value of orbits on return indicates that the orbits of the group are {0, 5, 7}, {1, 3, 6}, {2} and {4}.

Example 3:

1 3

6

7

g =

8

11

0

2

4 5

9 10

options[getcanon = TRUE, digraph = FALSE, writeautoms = TRUE, writemarkers =

TRUE, defaultptn = TRUE, tc level = 0].

output:

(8 11)(9 10) level 6: 10 orbits; 8 fixed; index 2

(7 8)(9 11) level 5: 8 orbits; 7 fixed; index 5

(4 6) level 4: 7 orbits; 4 fixed; index 2

(3 4)(5 6) level 3: 4 cells; 5 orbits; 3 fixed; index 4/9

(1 2) level 2: 3 cells; 4 orbits; 1 fixed; index 2

(0 1) level 1: 1 cell; 3 orbits; 0 fixed; index 3/12 orbits = (0,0,0,3,3,3,3,7,7,7,7,7), stats[grpsize1 = 480.0, grpsize2 = 0, numorbits = 3, numgenerators = 6, numnodes = 40, numbadleaves = 2, maxlevel = 7], lab = (3,4,6,5,7,8,11,9,10,0,1,2).

g =

9

10

0

11 1

2

5

4

3

7 8

6

Example 4: g =

11

6 7

4

3

2

9

10

5

8

0 1

options[getcanon = TRUE, digraph = FALSE, writeautoms = FALSE, writemarkers =

FALSE, defaultptn = TRUE, tc level = 0].

No output written.

14

orbits = (0,0,0,0,0,5,5,5,5,9,9,9), stats[grpsize1 = 480.0, grpsize2 = 0, numorbits = 3, numgenerators = 6, numnodes = 41, numbadleaves = 3, maxlevel = 7], lab = (5,6,8,7,0,1,4,2,3,9,10,11).

11 1 3

6

8

g =

9

10

0 2 4 5

which is identical to the resulting canong in Example 3.

7

8 User-defined procedures.

Provision is made for up to four procedures specified by the user to be called at various times during the processing. This will be done if pointers to them are passed in the userrefproc, userautomproc and/or usernodeproc, userlevelproc fields of options (see

Section

5 ). In all cases, a value of NULL will result in sensible default action.

The usertcellproc procedure disappeared in version 2.4. Its functionality can be reproduced (and exceeded) using the targetcell field of options.dispatch.

These procedures have many parameters in common; we will describe the most important of these here. Unless the individual procedure descriptions specify otherwise, they should be treated as read-only.

graph ∗g ; int m, n:

These are the arguments of the same name passed to nauty.

nauty has not changed them. See Section

5

for their meanings.

int level :

The level of the current node. The root of the search tree has level one.

int ∗lab, ∗ptn:

Arrays of length n giving partitions associated with each of the nodes along the path from the root of the tree to the current node. These are the parameters of the same name passed to nauty, but nauty has modified their contents as described below.

Suppose that we are currently at level l of the search tree. Let ν

1

, ν

2

, . . . , ν l path in the tree from the root ν

1 be the to the current node ν l

. The “partition at level i” is a partition π i associated with node ν i

. The partition originally passed to nauty, implicitly or explicitly, is the “partition at level 0”, denoted by π

0

. The complete partition nest

π

0

, π

1

, . . . , π l is held in lab and ptn thus:

(a) lab holds a permutation of {0, 1, . . . , n−1}.

(b) For 0 ≤ t ≤ l, the partition π t has as cells all the sets of the form {lab[i], lab[i+1],

. . . , lab[j]}, where [i, j] is a maximal subinterval of [0, n−1] such that ptn[k] > t for i ≤ k < j and ptn[j] ≤ t.

(c) Every entry of ptn which is not less than or equal to l is equal to NAUTY INFINITY.

(NAUTY INFINITY is a large constant defined in nauty.h.)

π

2

For example, say n = 10, l = 3, π

0

= [0, 2, 4, 6|8|5, 7, 9|3|1], and π ptn may be

3

= [0, 2, 4, 5, 6, 7, 8, 9|1, 3], π

1

= [0, 2, 4, 6|5, 7, 8, 9|1, 3],

= [4, 6|0, 2|8|5, 7, 9|3|1]. Then the contents of lab and

15

lab:

4 6 2 0 8 7 5 9 3 1 ptn:

3

1 2

∞ ∞

0 2 0

The order of the vertices within the cells of π l is arbitrary.

We will refer to the partition at level l as “the current partition”.

(a) userrefproc (g, lab, ptn, level, numcells, count, active, code, m, n)

This is a procedure to replace the default partition-refinement procedure, and is called for each node of the tree. The partition associated with the node is the “partition at level level ”, which is defined above.

The parameters passed are as follows.

g,m,n,lab,ptn,level : As above. The parameters lab and ptn may be altered by this procedure to the extent of making the current partition finer. The partitions at higher levels must not be altered.

int ∗numcells:

The number of cells in the current partition. This must be updated if the number of cells is increased.

permutation ∗count :

This is the address of an array of length at least n which can be used as scratch space. It can be changed at will.

set ∗active:

The set of active cells. This is not the same as the parameter of the same name passed to nauty, but has the same meaning and purpose. It can be changed without affecting nauty behaviour. See Section

5 .

int ∗code:

This must be set to a labelling-independent value which is an invariant of the partition at this level before or after refinement. (Example: the number of cells.)

It is essential that equivalent nodes have the same code. The value assigned must be less than NAUTY INFINITY.

The operation of refining the current partition involves permuting the vertices (i.e., entries of lab) within a cell, and then breaking it into subcells by changing the appropriate entries of ptn to level.

The validity of nauty requires that the operation performed be entirely independent of the labelling of the graph. Thus, if userrefproc is called with g and lab relabelled consistently and the same values of ptn and active, then the final values of ptn and active should be the same, and the final value of lab should be the same but relabelled in the same way (remembering always that the order of vertices within the cells doesn’t matter). It is also necessary that nodes of the tree which may be equivalent must be treated equivalently.

To be safe, regard any nodes on the same level as possibly equivalent.

It is desirable (but not compulsory) that the partition returned is equitable. If necessary, this can be done by calling the default refinement procedure refine, which has the same parameter list. If equitablility cannot be ensured, make sure that options.digraph =

TRUE.

The usefulness of userrefproc has declined since vertex-invariants were introduced (see

Section

9 ).

16

(b) usernodeproc (g, lab, ptn, level, numcells, tc, code, m, n)

This is called once for every node of the tree, after the partition has been refined.

The parameters passed are as follows. Treat all of them as read-only.

g,m,n,lab,ptn,level : As above.

int numcells:

The number of cells in the current partition.

int tc:

If nauty has determined that children of this node need to be explored, tc is the index in lab of where the target cell starts. Otherwise, it is −1.

int code:

This is the code produced by the refinement and vertex-invariant procedures while refining this partition.

(c) userautomproc(count , perm, orbits, numorbits, stabvertex , n)

This is called once for each generator of the automorphism group, in the same order as they are written (see Section

6 ). It is provided to facilitate such tasks as storing the

generators for later use, writing them in some unusual manner, or converting them into another representation (for example, into their actions on the edges).

Suppose the generator is γ = γ i

(j)

, in the notation of Section

6 . Then the parameters

have meanings as below. Treat them all as read-only.

int count :

The ordinal of this generator. The first is number 1.

permutation ∗perm:

The generator γ itself. For 0 ≤ i < n, perm[i] = i

γ

.

int ∗orbits; int numorbits:

The orbits and number of orbits of the group generated by all the generators found so far, including this one. See Section

5

for the format of orbits.

int stabvertex :

The value v j

, as defined in Section

6 .

int n:

The number of vertices, as usual.

(d) userlevelproc(lab, ptn, level, orbits, stats, tv, index, tcellsize, numcells, childcount, n)

This is called once for each node on the leftmost path downwards from the root, in bottom to top order. It corresponds to the markers “level ...”, which are described in Section

6 , except that an additional, initial, call is made for the first leaf of the tree.

The purpose is to provide more information than is provided by the markers, in a manner which enables it to be stored for later use, etc.. The parameters passed are as follows.

Treat them all as read-only.

n,lab,ptn,level : As above. The values of level will decrease by one for each call, reaching one for the final call.

Suppose that the value of level is l.

int ∗orbits:

The orbits of the group generated by all the automorphisms found so far.

See Section

5

for the format. In the notation of Section

6 , orbits gives the orbits of

the stabiliser Γ v

1

,v

2

,...,v l−1

.

statsblk ∗stats:

The meaning is as given in Section

5 , except that it applies to the

group generated by all the automorphisms found so far, that is to Γ v

1

,v

2

,...,v l−1

. Only

17

the fields which refer to the group can be assumed correct.

int tv , index , tcellsize, numcells:

In the notation of Section

6 , these are the values of v

l

, i l

, j l and c l

, respectively. For the first call, their values are 0, 1, 1 and n, respectively.

int childcount :

This is the number of children of the node at level level on the first path down the tree which were actually generated.

The condition numcells = n can be used to identify the first call.

9 Vertex-invariants.

As described in Section

2 , the operation of nauty is driven by a procedure which accepts

partitions and attempts to make them strictly finer without separating equivalent vertices.

For some families of difficult graphs, the built-in refinement procedure is insufficiently powerful, resulting in excessively large search trees. In many cases, this problem can be dramatically reduced by using some sort of invariant to assist the refinement procedure.

Formally, let G be the set of all labelled graphs (or digraphs) with vertex set V =

{0, 1, . . . , n−1}, and let Π be the set of partitions of V . As always, the order of the cells of a partition is significant, but the order of the elements of the cells is not. Let Z be the integers. A vertex-invariant is defined to be a mapping

φ : G × Π × V → Z such that φ(G

γ

, π

γ

, v

γ

) = φ(G, π, v) for every G ∈ G, π ∈ Π , v ∈ V and permutation γ.

Informally, this says that the values of φ are independent of the labelling of G.

A great number of vertex-invariants have been proposed in the literature, but very few of them are suitable for use with nauty. Most of them are either insufficiently powerful or require excessive amounts of time or space to compute. Even amongst the vertexinvariants which are known to be useful, their usefulness varies so much with the type of graph they are applied to, or the levels of the search tree at which they are applied, that intelligent automatic selection of a vertex-invariant by nauty would seem to be a task beyond our current capabilities. Consequently, the choice of vertex-invariant (or the choice not to use one) has been left up to the user.

The options parameter of nauty has four fields relevant to vertex-invariants, namely invarproc, mininvarlevel, maxinvarlevel and invararg. These are fully described in Section

5 . The I command in dreadnaut may be useful in investigating which of the supplied

vertex-invariants are useful for your problem. Experience shows that it is nearly always best to apply the invariant at just one level in the search tree, with levels 1 and 2 being the most likely candidates.

We now describe the vertex-invariants which are provided with nauty. Information on how to write a new vertex-invariant procedure can be found in the file nautinv.c. We will assume that g is a graph on V = {0, 1, . . . , n−1}, and that π = (V

0

, V

1

, . . . , V k

) is a partition of V . This partition will be equitable unless options.digraph = TRUE. One of the cells of π will be designated V

. If the procedure is called by nauty at level 1 (i.e.

at the root of the search tree), or directly by dreadnaut (I command), this will be the

18

first cell V

0

; otherwise, V

∗ will be the singleton cell containing the vertex fixed in order to create this node from its parent.

Unless otherwise specified, these invariants are only available for graphs in packed form. Trying to use them with sparse form will cause a disaster.

twopaths.

Each vertex v is given a code depending on the cells to which belong the vertices reachable from v along a path of length 2. invararg is not used. This is a cheap invariant suitable for graphs which are regular but otherwise have no particular structure

(for example).

adjtriang. Each vertex v is given a code depending on the number of common neighbours between each pair {v

1

, v

2

} of neighbours of v, and which cells v

1 and v

2 belong to. v

1 must be adjacent to v

2 if invararg = 0 and not adjacent if invararg = 1 . This is a fairly cheap invariant which can often break up the vertex sets of strongly-regular graphs.

triples. Each vertex v is given a code depending on the set of weights w(v, v

1

, v

2

), where

{v

1

, v

2

} ranges over the set of all pairs of vertices distinct from v such that at least one of

{v, v

1

, v

2

} lies in V

. The weight w(v, v

1

, v

2

) depends on the number of vertices adjacent to an odd number of {v, v

1

, v

2

} and to the cells that v, v

1 and v

2 belong to. invararg is not used. This invariant often works on strongly-regular graphs that adjtriang fails on, but is more expensive.

quadruples. Each vertex v is given a code depending on the set of weights w(v, v

1

, v

2

, v

3

), where {v

1

, v

2

, v

3

} ranges over the set of all pairs of vertices distinct from v such that at least one of {v, v

1

, v

2

, v

3

} lies in V

. The weight w(v, v

1

, v

2

, v

3

) depends on the number of vertices adjacent to an odd number of {v, v

1

, v

2

, v

3

} and to the cells that v, v

1

, v

2 and v

3 belong to. invararg is not used. This is an expensive invariant which can sometimes be of use for graphs with a particularly regular structure.

celltrips. Each vertex v is given a code depending on the set of weights w(v, v

1

, v

2

), where w(v, v

1

, v

2

) depends on the number of vertices adjacent to an odd number of {v, v

1

, v

2

}.

These three vertices are constrained to belong to the same cell. The cells of π are tried in increasing order of size until one splits. invararg is not used. This invariant can sometimes split the bipartite graphs derived from block designs, and other graphs of moderate difficulty.

cellquads.

Each vertex v is given a code depending on the set of weights w(v, v

1

, v

2

, v

3

), where w(v, v

1

, v

2

, v

3

) depends on the number of vertices adjacent to an odd number of

{v, v

1

, v

2

, v

3

}. These four vertices are constrained to belong to the same cell. The cells of

π are tried in increasing order of size until one splits. invararg is not used. This invariant is powerful enough to split many difficult graphs, such as hadamard-matrix graphs (where it is best applied at level 2).

cellquins. Each vertex v is given a code depending on the set of weights w(v, v

1

, v

2

, v

3

, v

4

), where w(v, v

1

, v

2

, v

3

, v

4

) depends on the number of vertices adjacent to an odd number of

{v, v

1

, v

2

, v

3

, v

4

}. These five vertices are constrained to belong to the same cell. The cells of π are tried in increasing order of size until one splits. invararg is not used. We know of no good use for this very powerful but very expensive invariant.

distances.

Each vertex v is given a code depending on the number of vertices at each

19

distance from v, and what cells they belong to. If a cell is found that splits, no further cells are tried. invararg specifies an upper bound on which distance to investigate, with 0 indicating no limit. This is a fairly cheap invariant suitable for things like regular graphs for which twopaths doesn’t work. Use the smallest value of invararg that works.

distances sg.

This is like distances but works for sparse form rather than packed form.

It is in the file nausparse.c rather than nautyinv.c.

indsets.

Each vertex v is given a code depending on the number of independent sets of size invararg which include v, and the cells containing the other vertices of those sets. The value of invararg is limited to 10. This can often split the vertex sets of strongly-regular graphs and bipartite design graphs, though it becomes expensive if invararg is large. A value of 4 is sometimes sufficient.

cliques. Each vertex v is given a code depending on the number of cliques of size invararg which include v, and the cells containing the other vertices of those cliques. The value of invararg is limited to 10. This can often split the vertex sets of strongly-regular graphs, though it becomes expensive if invararg is large. A value of 4 is sometimes sufficient.

cellcliq. Each vertex v is given a code depending on the number of cliques of size invararg which include v and lie within the cell containing v. The value of invararg is limited to 10.

The cells are tried in increasing order of size, and the process stops as soon as a cell splits.

This invariant applied at level 2 can be very successful on difficult vertex-transitive graphs.

A value of 3 can sometimes work even on strongly-regular graphs.

cellind.

Each vertex v is given a code depending on the number of independent sets of size invararg which include v and lie within the cell containing v. The value of invararg is limited to 10. The cells are tried in increasing order of size, and the process stops as soon as a cell splits. This invariant applied at level 2 can be very successful on difficult vertex-transitive graphs.

adjacencies. This is an invariant for digraphs and is not useful for graphs. The standard refinement procedure alone can sometimes give very poor performance for directed graphs, especially those which are not strongly connected. This invariant tries to correct the poor behaviour. Applying it to multiple levels may be necessary.

adjacencies sg.

This is like adjacencies but works for sparse form rather than packed form. It is in the file nausparse.c rather than nautyinv.c.

cellfano.

This invariant is intended for projective plane graphs but can be applied to any graphs. It is very expensive.

cellfano2.

This invariant is intended for projective plane graphs but can be applied to any graphs. It is very expensive, but maybe less than cellfano for genuine projective plane graphs. In the latter case, it can be thought of as counting the Fano subplanes according to which cells they involve. Another class of graph that this invariant can help with is the graphs derived from Latin squares as in Section

12 .

20

10 Writing programs which call nauty.

Programs which call nauty should include the file nauty.h. As well as defining the relevant types and parameters, this file also declares macros and procedures which are of use in constructing the arguments, and declares some useful tables.

In this section we will focus on packed graph representation, leaving sparse representation to Section

11 . Packed graph use of nauty requires the file naugraph.c. If a built-in

invariant is used, the file nautinv.c is also required.

Suppose that m and n have meanings as usual.

There are two general approaches to storage management. The first, the simplest if a prior limit is known on the graph size, is to define MAXN to be that limit before nauty.h

is included. nauty.h will define MAXM, and then MAXN and MAXN can be used to declare variables. For example: set s[MAXM]; /* a set */ graph g [MAXN*MAXM]; /* a graph */ int xy [MAXN]; /* an array */

The second method is more complicated but does not require a prior bound on the graph size. In this method, each variable whose size is unknown is dynamically allocated.

Of course you can do this yourself using malloc( ) but nauty.h provides macros for doing it in a convenient and efficient way. First there are static declarations:

DYNALLSTAT(set,s,s sz );

DYNALLSTAT(graph,g,g sz );

DYNALLSTAT(int,xy,xy sz );

Before the variables are used, they are set to the right size using the dynamic allocation macros:

DYNALLOC1(set,s,s sz,m,"malloc");

DYNALLOC2(graph,g,g sz,m,n,"malloc");

DYNALLOC1(int,xy,xy sz,n,"malloc");

To take the first variable as an example, the result of the macro will be that s has a value of type set∗ which points to an array of length at least m. If DYNALLOC1 or

DYNALLOC2 is used again for the same variable, it is freed and allocated again only if the new requested size is larger than the previous size. Otherwise the same space is reused. This is intended to be more efficient that repeated unnecessary calls to malloc( ) and free( ). In case it is desired to free the object allocated by DYNALLOC1, use, for example, DYNFREE(s,s sz ). There is also CONDYNFREE that frees objects if they are bigger than a given size.

In the case of g, we used DYNALLOC2 instead of DYNALLOC1. This is slightly better as it covers the possibility that mn is too large for an int. We could also use

DYNALLOC1(graph,g,g sz,m*(size t)n,"malloc");

The last parameter of DYNALLOC1 and DYNALLOC2 is a string used in an error message in the event that the allocation fails.

21

nauty.h also defines a number of macros that are useful for programming with the nauty data structures. Some of the more useful macros are as follows.

ADDELEMENT(s,i ) : add element i to set s.

DELELEMENT(s,i ) : delete element i from set s.

FLIPELEMENT(s,i ) : delete element i from set s if it is present, or insert it if it is absent.

ISELEMENT(s,i ) : test if i is an element of the set s (0 ≤ i ≤ n−1).

EMPTYSET(s,m) : make the set s equal to the empty set.

POPCOUNT(x ) : the number of 1-bits in the setword x.

Use (x ?POPCOUNT(x ):0) in circumstances where x is most often zero.

FIRSTBIT(x ) : the position (0 to WORDSIZE − 1) of the first (least-numbered) 1-bit in the setword x , or WORDSIZE if there is none.

TAKEBIT(i,x ) : If the setword x is not 0, set i to the position (0 to WORDSIZE − 1) of the first (least-numbered) 1-bit in x, and remove that bit from x.

ALLBITS : A setword constant with the first WORDSIZE bits set (this is usually all the bits).

BITMASK(i ) : A setword constant with the first i+1 bits unset and the other WORDSIZE− i − 1 numbered bits set, for 0 ≤ i < WORDSIZE. Thus, ANDing a setword with

BITMASK(i ) deletes bits 0..i.

ALLMASK(i ) : A setword constant with the first i bits set and all other bits unset, for

0 ≤ i ≤ WORDSIZE.

Some of the procedures in nautil.c or naugraph.c may be useful. They are declared in nauty.h. See the source code for the parameter list and semantics of these: nextelement : find the position of the next element in a set following a specified position.

The recommended way to do something for each element of the set s is like this: for (i = -1; (i = nextelement(s,m,i)) >= 0;)

{Process element i } permset : apply a permutation to a set.

orbjoin : update the orbits of a group according to a new generator.

writeperm : write a permutation to a file.

isautom : test if a permutation is an automorphism.

updatecan : (for samerows = 0) relabel a graph.

refine : find coarsest equitable partition not coarser than given partition.

refine1 : produces exactly the same results as refine, but assumes m = 1 for greater speed.

22

The file naututil.c contains procedures which are used by the dreadnaut program

(see Section

15 ). Many of these are also useful to programs which call nauty. If your

program uses them, include naututil.h instead of nauty.h.

Some of the more useful procedures are: setsize : find cardinality of set.

setinter : find cardinality of intersection of two sets.

putset : write a set to a file.

putgraph : write a graph to a file.

putorbits : write a set of orbits to a file.

putptn : write a partition to a file.

readgraph : read a graph from a file.

readptn : read a partition from a file.

ranperm : generate a random permutation.

rangraph : generate a random graph.

mathon : perform a doubling operation, as defined in [ 8 ].

complement : take the complement of a graph.

converse : take the converse of a digraph.

cellstarts : find the places where the cells at a given level begin.

sublabel : extract an induced subgraph of a graph.

In addition, the file nautaux.c contains a few procedures which manipulate graphs or partitions, but which are not currently used by nauty or dreadnaut.

It is recommended that programs which call nauty use the call nauty check (WORDSIZE,m,n,NAUTYVERSIONID); which will verify that a compatible verion of nauty is being used.

11 Programming with sparse representation.

The basic data structure for sparse representation is the structure sparsegraph defined in Section

3 . Programs using it should include nausparse.h and link with the

file nausparse.c.

As described in Section

3 , the sparse representation of a graph uses a structure of type

sparsegraph with the following fields: int nv : the number of vertices int nde: the number of directed edges (loops count as 1) int ∗v : pointer to an array of length at least nv

23

int ∗d : pointer to an array of length at least nv int ∗e: pointer to an array of length at least nde

SG WEIGHT ∗w : not implemented in this version, should be NULL size t vlen, dlen, elen, wlen: the actual lengths of the arrays v, d, e and w. The unit is the element type of the array in each case (so vlen is the number of ints in the array v, etc.)

For definiteness we will assume that such a graph is declared thus: sparsegraph sg;

Before use this should be initialised, for which there is a macro:

SG INIT(sg); or alternatively you can declare and initialise it at once:

SG DECL(sg);

To allocate the v, d and e arrays for a graph with n vertices and e directed edge, use

SG ALLOC(sg,n,e,"message"); where the message is used if allocation fails, and to free this space use

SG FREE(sg);

Many of the nauty procedures have a parameter of type graph∗, but you need to pass a pointer to your sparse graph structure. To do this, cast the pointer to the required type: (graph∗)(&sg).

A particular graph can be stored in several different ways, since the lists of neighbours of vertex do not need to be contiguous in sg.e, nor do they need to be sorted. To tell of two sparse graphs are identical, there is a procedure aresame sg in nausparse.c.

The canonically labelled graph produced by nauty is guaranteed to already have sorted contiguous adjacency lists. It also has a specific value of sg.v [i] if vertex i has degree 0, namely 0 for i = 0 and sg.v [i−1]+1 otherwise.

Some utilities for handling sparse form graphs can be found in nausparse.c: aresame sg : Test if two sparse graphs are the same. (Note: this is not an isomorphism test, just a labelled graph comparison.) sortlists sg : Sort the neighbourhood lists sg.e[sg.v [i] .. sg.v [i]+sg.v [i]-1] into ascending order.

put sg : Write a sparse graph in human-readable format.

sg to nauty : Convert sparse form to packed form.

nauty to sg : Convert packed form to sparse form.

distances sg : An invariant, see Section

9 .

24

12 Variations.

As mentioned, nauty can handle graphs with coloured vertices. In this section, we describe how several other types of isomorphism problem can be solved by mapping them onto a problem for vertex-coloured graphs.

Isomorphism of edge-coloured graphs.

Isomorphism of two graphs, each with both vertices and edges coloured, is defined in the obvious way. An example of such a graph appears at the left of Figure

3 .

4

1

2

3 1 0 0

0 0 3 0

2 0 0 2

2 0 1 0

3

1 2

Figure 3: Graphs with coloured edges

3

4

In the center of the figure we identify the colours with the integers 1, 2, 3. At the right of the figure we show an equivalent vertex-coloured graph. In this case there are two layers, each with its own colour. Edges of colour 1 are represented as an edge in the first (lowest) layer, edges of colour 2 are represented as an edge in the second layer, and edges of colour 3 are represented as edges in both layers. It is now easy to see that the automorphism group of the new graph (precisely, its action on the first layer) is the automorphism group of the original graph. Moreover, the order in which a canonical labelling of the new graph labels the vertices of the first layer can be taken to be a canonical labelling of the original graph.

More generally, if the edge colours are integers in {1, 2, . . . , 2 d

− 1}, we make d layers, and the binary expansion of each colour number tells us which layers contain edges. The vertical threads (each corresponding to one vertex of the original graph) can be connected graph has O(n log k) vertices. This can be improved to O(n

√ log k) vertices by also using edges that are not horizontal, but this needs care.

Exchangeable vertex colours.

The vertex colours known to nauty are distinguishable: vertices can only be mapped onto vertices of the same colour. In some applications, entire colour classes can also be exchanged.

In the left side of Figure

4

is a graph with three exchangeable vertex colours. To process this problem with nauty, we recolour the vertices to be all the same, then indicate the original colour classes with additional vertices of a new colour, as in the graph on the right.

It is easy to see how this idea can be extended to allow some colours to be exchangeable and some not.

25

Figure 4: Graphs with exchangeable vertex colours.

Isomorphism of hypergraphs and designs.

A hypergraph is similar to an undirected graph except that the edges can be vertex sets of any size, not just of size 2. Such an structure is also called a design.

5

4

3

1

2

1 1 0 0 0

0 1 1 0 0

0 0 1 1 1

1

2

3

4

5

Figure 5: Hypergraph/design isomorphism as graph isomorphism

In the left of Figure

5

we see a hypergraph with 5 vertices, two edges of size 2, and one edge of size 3. On the right is an equivalent vertex-coloured graph. The vertices on the left, coloured with one colour, represent the hypergraph edges, while the edges on the right, coloured with a different colour, represent the hypergraph vertices. The edges of the graph indicate the hypergraph incidence (containment) relationship.

In the center of the figure, we show the edge-vertex incidence matrix. This can be any binary matrix at all, which prompts us to note that the problem under consideration is just that of determining 0-1 matrix equivalence under independent permutation of the rows and columns. By combining this idea with the previous construction, we can handle such an equivalence relation on the set of matrices with arbitrary entries.

Hadamard equivalence.

Two matrices over {−1, +1} are Hadamard-equivalent if one can be obtained from the other by permuting the rows, permuting the columns, and multiplying some of the rows and some of the columns by −1.

Suppose A = (a ij

) is a matrix over {−1, +1} of order m × n. Construct a graph G(A) with vertices v

1

, . . . , v m

, v

0

1

, . . . , v

0 m colour. Insert the edges are {v i

, w j of one colour, and w

} and {v i

0

, w

0 j

} if a ij

1

, . . . , w n

, w

0

1

= 1 and {v i

, . . . , w

, w

0 j

0 n of another

} and {v i

0

, w j

} if a ij

= −1. Figure

6

gives an example.

26

w

1 w

2 w

3 w

0

1 w

0

2 w

0

3

1

−1

1

1

−1

1

−1 −1

1

 v

1 v

2 v

3 v

1

0 v

0

2 v

Figure 6: Hadamard equivalence as graph isomorphism

0

3

Permuting the rows of A corresponds to permuting v

1

, . . . , v m and v

1

0

, . . . , v

0 m together, and similarly for permuting the columns. Multiplying row i by −1 corresponds to interchanging v i with v

0 i

, and similarly with columns. Thus, the operations that define

Hadamard equivalence map onto graph isomorphism operations. It is less obvious that the same holds in reverse: if B is a second matrix, G(A) is isomorphic to G(B) if and

only if A is Hadamard-equivalent to B [ 3 ]. Similarly, Aut(G(A)) consists of the opera-

tions corresponding the Hadarmard equivalences of A to itself, together with the central element (v

1 v

0

1

) · · · (v m v

0 m

)(w

1 w

0

1

) · · · (w n w

0 n

) and a canonical labelling of G(A) can be used to make one of A. We omit the details.

Isotopy of matrices.

Two matrices over some symbol set S are called isotopic if one can be obtained from the other by permuting the rows, permuting the columns, and permuting the symbols. This equivalence relation is important in the study of Latin squares, quasigroups, and other subjects.

c

1 c

2 c

3 r

1

1 3 2

2 1 3

3 2 1 r

2 r

3 s

3 s

2 s

1

Figure 7: Isotopy as graph isomorphism

Figure

7

shows how to translate isotopy into isomorphism. There are four types of vertex, with four corresponding colours: one vertex for each row, one vertex for each column, one vertex for each symbol, and one vertex for each matrix position. The edges indicate in an obvious fashion what the row, column, and symbol is for each matrix

27

entry. Other related equivalences, such as paratopy (main class isotopy) can be handled

in similar fashion [ 5 ].

13 Installing nauty and dreadnaut .

First, read the file README to see if there is information more recent than this manual.

There are a number of source files provided. nauty by itself requires at least the files nauty.h, nauty.c, nautil.c and naugraph.c. The provided invariants are in nautinv.c.

The dreadnaut program requires, in addition, files rng.c, naututil.h, naututil.c and dreadnaut.c.

Starting at version 2.1, nauty does not try too hard to support very old or broken compilers. In particular, the basic facilities of ANSI C such as void and function prototypes are assumed. The files of nauty itself should compile with any ANSI-compliant compiler. On the other hand, dreadnaut expects a few library functions that might not be available in non-Unix systems.

If are using a Unix-like system (one that has a working shell and preferably make), the preferred way to compile nauty is to run the shell script

./configure

This will examine your system and create the files nauty.h, naututil.h, gtools.h and makefile in a way that is (hopefully) compatible. Then you can compile nauty using make nauty

If you are on a system where these tools are not available, but you have a halfdecent C compiler, you should start by editing the definitions near the start of nauty.h, naututil.h and gtools.h. (Most should be OK already.) Then you can compile using the commands in makefile as a guide. If you have trouble, please advise the author of the details. Similarly, please tell us if you can improve the operation on non-Unix systems.

This procedure will create two editions of dreadnaut, namely dreadnaut. The makefile also knows how to make other variants of nauty; here is a list of all of them: nauty.o etc: MAXN = 0 nauty1.o etc: MAXN = WORDSIZE (n limited to WORDSIZE) nautyW.o etc.: MAXN = 0, WORDSIZE = 32 nautyW1.o etc.: MAXN = WORDSIZE = 32 nautyS.o etc.: MAXN = 0, WORDSIZE = 16 nautyS1.o etc.: MAXN = WORDSIZE = 16 nautyL.o etc.: MAXN = 0, WORDSIZE = 64 nautyL1.o etc: MAXN = WORDSIZE = 64

The last two variants can only be used if your compiler supports either unsigned long or unsigned long long type as a 64-bit integer.

28

There are some test files included in the package. To run these, just use make checks which should not produce any output that looks like an error message.

14 Efficiency.

The theoretical worst-case efficiency of nauty is exponential, has proved by Miyazaki [ 6 ].

However, the worst cases are rather hard to find and for most classes of graphs nauty behaves as if it is polynomial.

Here we give some sample execution times for a Pentium III processor at 1 GHz, using gcc with full optimisation, using default options and packed form unless otherwise specified.

For random graphs with edge probability about n

2

1

, average execution times for large n are

2 nanoseconds with options.getcanon = FALSE, and 12 n

2 nanoseconds with options.getcanon = TRUE. The large difference between these times for large n is almost entirely taken up by the process of permuting the entries of g to get canong. Except for very small n, nearly all random graphs have only discrete equitable partitions, and thus have trivial automorphism groups. All nauty does in this case is one refinement operation followed, if options.getcanon = TRUE, by one relabelling operation.

The 46308 vertex-transitive graphs of order 30 require 0.14 milliseconds each on average, irrespective of the value of options.getcanon. An average of about 4 generators each are found for the automorphism groups.

A list of difficult graphs is given by Mathon in [ 8 ]. Using his notation for them, we

find the following times with options.getcanon = TRUE:

A

25

–B

25

: 0.00015 seconds;

A

50 and B

50

: 0.029 seconds (0.006 seconds using vertex-invariant cellquads at level 1);

A

1

25

: 0.0008 seconds (0.00014 seconds using vertex-invariant adjtriang at level 1);

B

1

25

: 0.0026 seconds (0.00014 seconds using vertex-invariant adjtriang at level 1);

A

35

–D

35

: 0.008 seconds (0.00018 seconds using vertex-invariant cliques with parameter 4 at level 1);

A

52

: 0.009 seconds (0.0014 seconds using vertex-invariant cliques with parameter 5 at level 1);

B

52

: 0.055 seconds (0.0016 seconds using vertex-invariant cliques with parameter 5 at level 1);

A

72

–D

72

: 0.58 seconds (0.15 seconds using vertex-invariant quadruples applied at level 1).

The execution time for these graphs varies somewhat with the initial labelling.

Amongst the most difficult known graphs for this algorithm, and probably for most other similar algorithms, are certain bipartite graphs derived from Hadamard matrices.

29

For example, some of these graphs on 96 vertices require more than 5 seconds to process.

However, with the vertex-invariant cellquads applied at level two, this time is reduced to less than 1 second.

A family of strongly-regular graphs with 155 vertices and trivial automorphism group require 0.76 seconds with no vertex-invariant and 0.01 seconds with the vertex-invariant adjtriang (or cliques with parameter 4) applied at level 1. A similar family with 1027 vertices require 264 seconds with no vertex-invariant and 1.3 seconds with the vertexinvariant adjtriang (or cliques with parameter 4) applied at level 1.

As examples of how nauty performs for very rich automorphism groups, we mention

L(K

30

) (435-vertex linegraph of complete graph; group size 30!; execution time 0.1 seconds; 29 generators) and the 1-skeleton of the 11-cube (2048-vertex graph; group size

81,749,606,400; execution time 11 seconds; 11 generators).

The space efficiency of sparse form is better than packed form if the number of directed edges is less than n

2

/32, approximately.

The time efficiency of sparse form is also, not surprisingly, better than packed form if the graph is sparse. Applications need to be assessed individually but, as a rough guide, sparse form is better for n vertices when the average degree is less than or equal to d, according to the following table.

n 50 100 500 1000 5000 10000 50000 d 4 6 16 24 65 109 240

In the following table we give times for random cubic graphs of various sizes. The line labelled “invar” is for sparse form using the invariant distances sg with parameter 4.

n 20 50 100 500 1000 5000 10000 20000 dense 0.20 ms 2.4 ms 16 ms 2.1 s 20 s sparse 0.19 ms 1.4 ms 6.0 ms 0.24 s 1.2 s 96 s invar 0.033 ms 0.15 ms 0.41 ms 6.9 ms 29 ms 1.3 s 10 s 67 s

Another example of sparse form efficiency is given by the d-dimensional cubes, which are regular of degree d and have n = 2 d vertices and group size 2 d d! . Compare the following to 11 seconds for d = 11 using packed form.

d n

11 12 14 seconds 0.02

0.07

0.41

16

2048 4096 16384 65536 262144 1048576 4194304

2.9

18

25

20

219

22

1628

30

15 dreadnaut .

dreadnaut is a simple program which can read graphs and execute nauty. It is only a very primitive interface with few facilities. If you want to use nauty in a richer interactive environment, some of your choices are:

(a) Magma: http://magma.maths.usyd.edu.au/magma

(b) GAP with GRAPE: http://www.maths.qmul.ac.uk/

∼ leonard/grape/

(c) LINK: http://dimacs.rutgers.edu/

∼ berryj/LINK.html

(d) Vega: http://www.ijp.si/vega/

(e) MuPAD with MuPAD-Combinat: http://mupad-combinat.sourceforge.net/

Input is taken from the standard input and output is sent to the standard output, but this can be changed by using the “<” and “>” commands. Commands may appear any number per line separated by white space, commas, semicolons or nothing. They consist of single characters, sometimes followed by parameters.

At any point of time, dreadnaut knows the following information:

(a) The number of vertices, n.

(b) The “current graph” g, if defined.

(c) The “current partition” π, if defined.

(d) The orbits of the (coloured) graph (g, π), if defined.

(e) The canonically labelled isomorph of g, called h, if defined. (Also called canong.)

(f) An extra graph called h

0

, if defined. (Also called savedg.)

(g) Values for each of a variety of options.

In the following ‘#’ is an integer and ‘=’ is optional.

(A)

Commands which define or examine the graph g.

n=#

Set value of n. The maximum value is installation-defined.

g

Read the graph g.

There is always a “current vertex” which is initially the first vertex. (Vertices are numbered from 0 unless you have used the $ command.) The number of the current vertex is displayed as part of the prompt, if any. Available subcommands:

#

: add an edge from the current vertex to the specified vertex. (Unless you have selected the option digraph, edges only need to be entered in one direction.)

-# : delete the edge, if any, from the current vertex to the specified vertex.

;

: increment the current vertex. If it becomes too high for a vertex label, stop.

#: : make the specified vertex the current vertex.

?

: display the neighbours of the current vertex.

.

: stop.

!

: ignore the rest of this input line.

,

: ignored.

e

Edit the graph g. The available subcommands are the same as for the “g” command.

r . . . ;

Relabel the graph g, where ‘. . .’ is a permutation of {0, 1, . . . , n−1}, specifying the order in which to relabel the vertices, followed by a semicolon. Missing numbers

31

are filled in at the end in numerical order. For example, for n = 5, “r 4,1;” is equivalent to “r 4,1,0,2,3;”. The partition π is permuted consistently.

R . . . ;

This is the same as r except that unspecified vertices are not filled in. Instead, a subgraph corresponding to the given vertices is formed and replaces g. If the command is given as -R, the given vertices are deleted instead. The partition π is reset to have only one cell.

j

%

Relabel the graph g at random. The partition π is permuted consistently.

Perform the doubling operation E(g) defined in [ 8 ]. The result in g is a regular

graph with order 2n + 2 and degree n.

s=#

Generate graph (or digraph) g at random with independent edge probabilities 1/i, where i is the integer specified.

(underscore) Replace the graph g by its complement. If there are any loops, the set of loops is complemented too; otherwise, no loops are introduced.

(two underscores) If g is a digraph, take its converse (which reverses the direction of all the edges). Otherwise do the same as .

t

T v

Type the graph g, in an obvious format. The value of option linelength is taken into account. The format used is consistent with the input format allowed by the “g” command. To examine just some of the graph, you can use the “?” subcommand within the “e” command.

This is exactly like “t” except that a line of the form “n=n $=l g” is written first, where n is the number of vertices and l is the number of the first vertex, and a line of the form “$$” is written afterwards. This enables you to save a graph to a file and easily restore it later: “>newgraph.dre T ->” will save g to the file newgraph.dre, while “<newgraph.dre” will restore it.

Display the degrees of each vertex of the graph g, if defined. For digraphs, the outdegrees are displayed.

(B) f i

I

Commands which define the partition π.

Specify an initial partition.

“-f” selects the partition with only one cell, which is the default.

“f=#” selects the partition with one cell containing just the vertex named and one cell containing every other vertex.

“f=[ . . . ]” selects an arbitrary partition. Replace “. . .” by a list of cells separated by “|”. You can use the abbreviation “x:y” for the range x, x+1, . . . , y. Any vertices not named are put in a cell of their own at the end.

Example: If n = 10, then “f=[3:7 | 0,2]” establishes the partition

[3, 4, 5, 6, 7 | 0, 2 | 1, 8, 9].

Perform a refinement operation, replacing the partition π by its refinement. The active set initially contains every cell.

Perform a refinement operation, an application of the vertex-invariant (if one has been selected using the * command), and (if any cells were split) another refinement

32

operation. The final partition becomes π. The behaviour may be modified by the

K command, but not by the k command.

This is useful for determining whether an invariant is effective for a particular graph.

Note that you need to restore the partition between repeated tests.

(C)

Commands which establish or examine options.

$=#

Establish an origin for vertex numbering. The default is 0. Only non-negative values are permitted. All the input-output routines used by nauty or dreadnaut respect this value, even though internally vertices are always numbered from 0.

(The value given is copied into the global int variable labelorg, which is described in Section

5 .)

$$

Restore the vertex numbering origin to what it was just before the last $ command.

Only one previous value is remembered.

l=#

Set value of option linelength : the length of the longest line permitted for output.

The default value is installation-dependent (typically 78).

w=#

Set value of worksize : the amount of space provided for nauty to store automorphism data. The maximum value is installation-defined, and the default is the same as the maximum. There’s little reason to ever use this command.

+

Ignored. Provided for contrast with “-”.

d,-d

Set option digraph to TRUE or FALSE, respectively. You must set it to TRUE if you wish to define g to be a digraph or a graph with loops. The default is FALSE.

Changing it from TRUE to FALSE also causes the graph g to become undefined, as a safety measure.

c,-c

Set option getcanon to TRUE or FALSE, respectively. This tells nauty whether to find a canonical labelling or just the automorphism group. The default is FALSE.

a,-a

Set option writeautoms to TRUE or FALSE, respectively. This tells nauty whether to display the automorphisms it finds. The default is TRUE.

m,-m

Set option writemarkers to TRUE or FALSE, respectively.

This tells nauty whether to display the level markers “level . . . ”. See Section

6

for their meaning.

The default is TRUE.

p,-p

Set option cartesian to TRUE or FALSE, respectively. This tells nauty to use the “cartesian” form when writing automorphisms. Precisely, the automorphism γ is displayed as a list v

γ

1 v

γ

2

. . . v

γ n

, where v

1

, v

2

, . . . , v n are the vertices of g. The default is FALSE.

y=#

Set the value of option tc level. A value of # tells nauty to use an advanced, but expensive, algorithm for choosing target cells in the top k levels of the search tree.

See Section

5

for a more detailed description. The default is 100, but setting it to

0 might speed up the average time for easy graphs.

∗=# Select a vertex-invariant. One user-defined vertex-invariant can be linked with dreadnaut if its name is provided in the preprocessor variable INVARPROC. The argument to the ∗ command is interpreted thus:

33

-1 : the user-defined procedure (if any)

0

: no vertex-invariant (this is the default)

1

: twopaths

2

: adjtriang

3

: triples

4

: quadruples

5

: celltrips

6

: cellquads

7

: cellquins

8

: distances

9

: indsets

10 : cliques

11 : cellcliq

12 : cellind

13 : adjacencies

14 : cellfano

15 : cellfano2

These procedures are described in Section

9 .

The default behaviour is for the invariant to be applied only at the root of the tree, but this can be modified using the k command. The K command can be used to change the invariant parameter, if there is one. The default is K=3 for indsets, cliques, cellind and cellcliq; and K=0 for everything else.

k=# #

(Two integer arguments.) Define values for the options mininvarlevel and maxinvarlevel. These tell nauty the minimum and maximum levels of the tree at which it is to apply the vertex-invariant. The root of the tree is at level 1. See Section

5

for a little more information about these options. The default is k = 0 1, which causes the invariant to be applied only at the top of the search tree.

K=#

Give a value to the invararg option. This number is passed to the vertex-invariant by the I command and by nauty. See Section

9

for the meaning of this option for each available vertex-invariant. The default value depends on the invariant; see the

∗ command.

u=#

Request calls to user-defined functions. The value is

1 for usernodeproc,

2 for userautomproc,

4 for userlevelproc,

16 for userrefproc.

These can be added together to select more than one procedure. The procedures called are those named by the compile-time symbols USERNODE, USERAUTOM,

USERLEVEL, USERTCELL and USERREF defined in dreadnaut.c. The default values are:

USERNODE: For each node, print a number of dots equal to the depth, then

(numcells /code /tc ) where numcells is the number of cells, code is the code produced by the refinement procedure, and tc is the position in lab where the target cell starts. For the first path down the tree, the partition is displayed as well.

34

?

USERAUTOM: For each automorphism, display the arguments numorbits and stabvertex (see Section

8 ).

USERLEVEL: For each level, display the arguments tv , index , tcellsize, numcells and childcount, as well as the fields numnodes, numorbits and numgenerators of stats. See Section

8

for what they mean.

USERREF: Do nothing.

Type the current values of m, n, worksize, most of the options, the number of edges in g, and the number of cells in π. If output has been directed away from stdout using the “>” command, some of this information is also written to stdout.

&

Type the current partition π, unless it is has only one cell.

&&

Same as &, except that the quotient of g with respect to π is also written. Say

π = (V

0

, V

1

, . . . , V m

) and let v i be the least numbered vertex in V i for 0 ≤ i ≤ m.

Then, for each i, this command writes v i

, then |V i

| in brackets, then the numbers k

0

, k

1

, . . . , k m

, where k j is the number of edges from v j as “-”, while the value |V i

| is written as “∗”.

to V i

. The value 0 is written

(D) x

Commands which execute nauty or use the results.

Execute nauty. Depending on the values of the writeautoms and writemarkers options, the automorphism group will be displayed while nauty is running. See

Section

6

for an explanation of the output. When nauty returns, dreadnaut will display some statistics about it. See Section

5

for the meanings; the important ones are the order of the group and the number of orbits. Depending on your system, the execution time is also displayed.

Copy h, if defined, to h

0

. See the description of the # command for more.

@ b

Type the canonical label and the canonically labelled graph. The canonical label is given in the form of a list of the vertices of g in canonical order. Only possible after x with option getcanon selected.

z

Type two 8-digit hex numbers whose value depends only on h. This allows quick comparison between graphs. Isomorphic graphs give the same value. In principle, non-isomorphic graphs may also give the same value but we don’t know any examples (please tell us if you find one). Only possible after x with option getcanon selected.

#

Compare the labelled graphs h and h

0

. Both must have been already defined (using x and @). The complete process for testing two graphs g

1 and g

2 for isomorphism is this: enter g

1 c x @

(select getcanon option, execute nauty, copy h to h

0

); enter g

2 x #

(execute nauty, compare h to h

0

).

##

This is the same as # except that, if h is identical to h

0

, you will also be given an isomorphism from g

1 to g

2

. This is in the form of a sequence of pairs v i

-w i

, where v i is a vertex of g

1 and w i is a vertex of g

2

. The vertex-numbering origin in force

35

when h

0 was created is used for g

1

, whilst the origin now in force is used for g

2

.

Type the orbits of the group. Only possible after x.

o

(F)

Miscellaneous commands.

h,H

Help: type a summary of dreadnaut commands.

"..."

Anything between the quotes is simply copied to the output. The ligatures ‘\n’

(newline), ‘\t’ (tab), ‘\b’ (backspace), ‘\r’ (carriage return), ‘\f’ (formfeed), ‘\\’

(backslash), ‘\’’ (single quote) and ‘\"’ (double quote) are recognised. Other occurrences of ‘\’ are ignored.

!

Ignore anything else on this input line. Note that this is a command, not a comment character in the usual sense, so you can’t use it in the middle of other commands.

<

Begin reading input from another file. The name of the file starts at the first nonwhite character after the “<” and ends before the next white character. If such a file cannot be found, another attempt is made with the string “.dre” appended to the name. When end-of-file is encountered on that file, continue from the current input file. The allowed level of nesting is system-dependent (usually 8).

>,>>

Close the existing output file unless it is the standard output, then begin writing output to another file. The name of the file starts at the first non-white character after the “>” and ends before the next white character. For “>” the file starts off empty. For “>>”, if an existing file of the right name exists, it is written to starting at the current end-of-file. Use “->” to direct output back to the standard output.

M=#

Each call to nauty is performed # times and the average cpu time is then reported accurately. This is for doing timing tests with easy graphs.

q

Quit. dreadnaut will exit irrespective of which level of input nesting it is on.

The canonical labellings produced by dreadnaut can depend on the values of many of the options. If you are testing two or more graphs for isomorphism, it is important that you use the same values of these options for all your graphs. In general, h is a function of all these:

(a) option digraph (d command)

(b) all the vertex-invariant options (∗, k and K commands)

(c) the value of tc level (y command)

(d) the use of userrefproc (u command)

(e) the version of nauty used

Beginning at version 2.1, the canonical labelling does not depend on the compiler, the system, or the word size.

36

Several sample dreadnaut sessions are shown below. The first problem solved is the second example in Section

7 . The underlined characters are those typed by the user.

> n=8 g

0: 1 3 4;

1: 2 5;

2: 3 6;

3: 7;

4: 5 7;

5: 6;

6: 7.

> f=2 x

[fixing partition]

8 vertices enter the graph fix vertex 2; execute

(0 5)(3 6) level 2: 6 orbits; 3 fixed; index 2

(1 3)(5 7) level 1: 4 orbits; 1 fixed; index 3

4 orbits; grpsize=6; 2 gens; 6 nodes; maxlev=3 tctotal=7; cpu time = 0.00 seconds

> o show the orbits

0 5 7; 1 3 6; 2; 4;

> q quit

The next problem solved is to determine an isomorphism between the graphs of examples 3 and 4 of Section

7 . We turn off the writing of automorphisms to save some

space.

> c -a -m

> n=12 g

0: 1; 2; 0; turn getcanon on, group writing off enter the first graph

3: 4; 5; 6; 3;

7: 8; 9; 10; 11; 7.

> x @ execute, save the result

3 orbits; grpsize=480; 6 gens; 31 nodes (3 bad leaves); maxlev=7 tctotal=88; canupdates=1; cpu time = 0.00 seconds

> g enter the second graph

0: 1; 2; 3; 4; 0;

5: 6; 7; 8; 5;

9: 10; 11; 9.

> x execute

3 orbits; grpsize=480; 6 gens; 50 nodes (2 bad leaves); maxlev=7 tctotal=124; canupdates=4; cpu time = 0.00 seconds

> ## compare to saved graph h and h’ are identical.

0-9 1-10 2-11 3-5 4-6 5-7 6-8 7-0 8-1 9-2 10-3 11-4

37

As a third example, we consider a simple block design. nauty can compute automorphisms and canonical labellings of block designs by the common method of converting the design to an equivalent coloured graph. Suppose a design D has varieties x

1

, x

2

,

. . ., x v and blocks B

1

, B

2

, . . ., B b

. Define G(D) to be the the graph with vertex set

{x

1

, . . . , x v

, B

1

, . . . , B b

}, with each x-vertex having one colour and each B-vertex having a second colour, and edge set {x i

B j

| x i

∈ B j

}. The following theorem is elementary.

Theorem 2.

(a) The automorphism group of D is isomorphic to the automorphism group of G(D).

(b) If D

1 and D

2 are designs, D

1

G(D

2

) are isomorphic.

and D

2 are isomorphic if and only if G(D

1

) and

Consider the design D = { {1, 2, 4}, {1, 3}, {2, 3, 4} }. Label G(D) so that the varieties of D correspond to vertices 1–4, while the blocks correspond to vertices 5–7.

> $=1

> n=7 g

1: 5:

5: 1 2 4;

6: 1 3; label vertices starting at 1 go to vertex 5 (block 1)

7: 2 3 4.

> f=[1:4]

> cx fix the varieties setwise run nauty

[fixing partition]

(2 4) level 2: 6 orbits; 2 fixed; index 2

(1 3)(5 7) group generators level 1: 4 orbits; 1 fixed; index 2

4 orbits; grpsize=4; 2 gens; 6 nodes; maxlev=3 tctotal=6; canupdates=1; cpu time = 0.00 seconds

> o display the orbits

1 3; 2 4; 5 7; 6;

> b

1 3 2 4 6 5 7

1 : 5 6;

2 : 5 7; display the canonical labelling the vertices in canonical order the relabelled graph

3 : 6 7;

4 : 6 7;

5 : 1 2;

6 : 1 3 4;

7 : 2 3 4;

> q quit

For many families of block designs, it can be proved that the isomorphism class of each design is uniquely determined by the isomorphism class of its block-intersection graph, where that graph has the blocks as vertices and pairs of intersecting blocks as edges. For

(v, k, 1)-designs, a sufficient condition for this is that v > k(k

2

− 2k + 2). On the occasions

38

when this is true, nauty can usually process the block-intersection graphs more quickly than it can process the designs directly. Also, the vertex-invariants described in Section

9

are more likely to be successful with the block-intersection graphs.

16 gtools.

The nauty package includes a suite of programs called gtools that provide efficient processing of files of graphs stored in graph6 or sparse6 format. These formats are defined in the file formats.txt. Support for gtools is limited to Unix, but they may run on other systems which provide basic Unix system calls. The program shortg requires a program compatible with the Unix sort program, as well as pipes.

All the gtools programs are self-documenting: just execute with the option --help to see an explanation of all the features. We only list the basic functions of the programs here; see Section

20

for more details.

geng : generate small graphs genbg : generate small bicoloured graphs gentourng : generate small tournaments directg : generate small digraphs with given underlying graph multig : generate small multigraphs with given underlying graph genrang : generate random graphs copyg : convert format and select subset labelg : canonically label graphs shortg : remove isomorphs from a file of graphs listg : display graphs in a variety of forms showg : a stand-alone version of listg amtog : read graphs in adjacency matrix form dretog : read graphs in dreadnaut form complg : complement graphs catg : concatenate files of graphs addedgeg : add an edge in each possible way deledgeg : delete an edge in each possible way newedgeg : in each possible way, subdivide two non-adjacent edges and join the two new vertices

NRswitch : switch the edges between the neighbourhood and the complementary neighbourhood, for each vertex countg : count graphs according to a variety of properties

39

pickg : select graphs according to a variety of properties biplabg : label bipartite graphs so the colour classes are contiguous planarg : test graphs for planarity and find embeddings or obstructions. The planarity

code uses the method of Boyer and Myrvold [ 1 ] and was programmed by Paulette

Lieby for the Magma project.

Further programs will be added. Requests are welcome.

17 Recent changes.

This section lists the most significant changes made to nauty or dreadnaut since version 2.2. For a complete list of even trivial changes, see the source code and the file

README.

(a) Sparse representation of graphs is now supported. There are new files nausparse.h

and nausparse.c. Only the program labelg uses sparse representation so far.

(b) New utilities planarg, multig and gentourng.

(c) The hook usertcellproc has been removed. The same and greater functionality can be achieved using options.dispatch.targetcell.

(d) New switches -R,-l,-m in genrang, -f,-V in directg, -i,-I,-K,-T in shortg, and -T in countg and pickg.

(e) BIGNAUTY has gone. Programs previously linked with object files like nautyB.o

can now be linked to nauty.o.

18 Sample programs which call nauty.

In this section we give five sample programs which illustrate the use of nauty.

The first two programs illustrate normal usage of nauty with graphs in packed form.

The first uses a fixed positive value of MAXN so is limited to that size. The second uses dynamic allocation and so works with much larger sizes.

The third program illustrates how the code in the files naugroup.h and naugroup.c

can be used to produce the full automorphism group of a graph one element at a time.

The fourth program illustrates basic usage of nauty with graphs in sparse form.

The fifth program illustrates how to use nauty to find an isomorphism between two graphs.

These programs are available in the nauty distribution and known to the makefile.

40

18.1

nautyex1.c : Packed form with static allocation

/* This program prints generators for the automorphism group of an n-vertex polygon, where n is a number supplied by the user.

It needs to be linked with nauty.c, nautil.c and naugraph.c.

*/

This version uses a fixed limit for MAXN.

#define MAXN 1000 /* Define this before including nauty.h */

#include "nauty.h" /* which includes <stdio.h> and other system files */ int main(int argc, char *argv[])

{ graph g[MAXN*MAXM]; int lab[MAXN],ptn[MAXN],orbits[MAXN]; static DEFAULTOPTIONS_GRAPH(options); statsblk stats; setword workspace[5*MAXM]; int n,m,v; set *gv;

/* Default options are set by the DEFAULTOPTIONS_GRAPH macro above.

* Here we change those options that we want to be different from the

* defaults.

writeautoms=TRUE causes automorphisms to be written.

*/ options.writeautoms = TRUE; while (1)

{ printf("\nenter n : "); if (scanf("%d",&n) != 1 || n <= 0) break;

/* Exit if EOF or bad number */ if (n > MAXN)

{ printf("n must be in the range 1..%d\n",MAXN); exit(1);

}

/* The nauty parameter m is a value such that an array of

* m setwords is sufficient to hold n bits.

The type setword

* is defined in nauty.h.

The number of bits in a setword is

* WORDSIZE, which is 16, 32 or 64.

Here we calculate

* m = ceiling(n/WORDSIZE).

*/

41

} m = (n + WORDSIZE - 1) / WORDSIZE;

/* The following optional call verifies that we are linking

* to compatible versions of the nauty routines.

*/ nauty_check(WORDSIZE,m,n,NAUTYVERSIONID);

/* Now we create the cycle.

For each v, we add the edges

* (v,v+1) and (v,v-1), where values are mod n.

gv is set to

* the position in g[] where row v starts, EMPTYSET zeros it,

* then ADDELEMENT adds one bit (a directed edge) to it.

*/ for (v = 0; v < n; ++v)

{ gv = GRAPHROW(g,v,m);

EMPTYSET(gv,m);

ADDELEMENT(gv,(v+n-1)%n);

ADDELEMENT(gv,(v+1)%n);

} printf("Generators for Aut(C[%d]):\n",n);

/* Since we are not requiring a canonical labelling, the last

* parameter to nauty() is noit required and can be NULL.

* Similarly, we are not using a fancy active[] value, so we

* can pass NULL to ask nauty() to use the default.

*/ nauty(g,lab,ptn,NULL,orbits,&options,&stats, workspace,5*MAXM,m,n,NULL);

/* The size of the group is returned in stats.grpsize1 and

* stats.grpsize2. See dreadnaut.c for code that will write the

* value in a sensible format; here we will take advantage of

* knowing that the size cannot be very large.

Adding 0.1 is

* just in case the floating value is truncated instead of rounded,

* but that shouldn’t be.

*/ printf("Automorphism group size = %.0f",stats.grpsize1+0.1);

} exit(0);

42

18.2

nautyex2.c : Packed form with dynamic allocation

/* This program prints generators for the automorphism group of an n-vertex polygon, where n is a number supplied by the user.

It needs to be linked with nauty.c, nautil.c and naugraph.c.

*/

This version uses dynamic allocation.

#include "nauty.h" /* which includes <stdio.h> */

/* MAXN=0 is defined by nauty.h, which implies dynamic allocation */ int main(int argc, char *argv[])

{

/* DYNALLSTAT declares a pointer variable (to hold an array when it

* is allocated) and a size variable to remember how big the array is.

* Nothing is allocated yet.

*/

DYNALLSTAT(graph,g,g_sz);

DYNALLSTAT(int,lab,lab_sz);

DYNALLSTAT(int,ptn,ptn_sz);

DYNALLSTAT(int,orbits,orbits_sz);

DYNALLSTAT(setword,workspace,workspace_sz); static DEFAULTOPTIONS_GRAPH(options); statsblk stats; int n,m,v; set *gv;

/* Default options are set by the DEFAULTOPTIONS_GRAPH macro above.

* Here we change those options that we want to be different from the

* defaults.

writeautoms=TRUE causes automorphisms to be written.

*/ options.writeautoms = TRUE; while (1)

{ printf("\nenter n : "); if (scanf("%d",&n) == 1 && n > 0)

{

/* The nauty parameter m is a value such that an array of

* m setwords is sufficient to hold n bits.

The type setword

* is defined in nauty.h.

The number of bits in a setword is

* WORDSIZE, which is 16, 32 or 64.

Here we calculate

* m = ceiling(n/WORDSIZE).

*/

43

} m = (n + WORDSIZE - 1) / WORDSIZE;

/* The following optional call verifies that we are linking

* to compatible versions of the nauty routines.

*/ nauty_check(WORDSIZE,m,n,NAUTYVERSIONID);

/* Now that we know how big the graph will be, we allocate

* space for the graph and the other arrays we need.

*/

DYNALLOC2(graph,g,g_sz,m,n,"malloc");

DYNALLOC1(setword,workspace,workspace_sz,5*m,"malloc");

DYNALLOC1(int,lab,lab_sz,n,"malloc");

DYNALLOC1(int,ptn,ptn_sz,n,"malloc");

DYNALLOC1(int,orbits,orbits_sz,n,"malloc"); for (v = 0; v < n; ++v)

{ gv = GRAPHROW(g,v,m);

EMPTYSET(gv,m);

ADDELEMENT(gv,(v+n-1)%n);

ADDELEMENT(gv,(v+1)%n);

} printf("Generators for Aut(C[%d]):\n",n); nauty(g,lab,ptn,NULL,orbits,&options,&stats, workspace,5*m,m,n,NULL);

/* The size of the group is returned in stats.grpsize1 and

* stats.grpsize2. See dreadnaut.c for code that will write the

* value in a sensible format; here we will take advantage of

* knowing that the size cannot be very large.

Adding 0.1 is

* just in case the floating value is truncated instead of rounded,

* but that shouldn’t be.

*/ printf("Automorphism group size = %.0f",stats.grpsize1+0.1);

} else break;

} exit(0);

44

18.3

nautyex3.c : Finding the whole automorphism group

/* This program prints the entire automorphism group of an n-vertex polygon, where n is a number supplied by the user.

It needs to be linked with nauty.c, nautil.c, naugraph.c and naugroup.c.

*/

#include "nauty.h"

#include "naugroup.h"

/* which includes <stdio.h> */

/**************************************************************************/ void writeautom(permutation *p, int n)

/* Called by allgroup.

Just writes the permutation p. */

{ int i; for (i = 0; i < n; ++i) printf(" %2d",p[i]); printf("\n");

}

/**************************************************************************/ int main(int argc, char *argv[])

{

DYNALLSTAT(graph,g,g_sz);

DYNALLSTAT(int,lab,lab_sz);

DYNALLSTAT(int,ptn,ptn_sz);

DYNALLSTAT(int,orbits,orbits_sz);

DYNALLSTAT(setword,workspace,workspace_sz); static DEFAULTOPTIONS_GRAPH(options); statsblk stats; int n,m,v; set *gv; grouprec *group;

/* The following cause nauty to call two procedures which store the group information as nauty runs. */ options.userautomproc = groupautomproc; options.userlevelproc = grouplevelproc; while (1)

{ printf("\nenter n : ");

45

} if (scanf("%d",&n) == 1 && n > 0)

{ m = (n + WORDSIZE - 1) / WORDSIZE; nauty_check(WORDSIZE,m,n,NAUTYVERSIONID);

DYNALLOC2(graph,g,g_sz,m,n,"malloc");

DYNALLOC1(int,lab,lab_sz,n,"malloc");

DYNALLOC1(int,ptn,ptn_sz,n,"malloc");

DYNALLOC1(int,orbits,orbits_sz,n,"malloc");

DYNALLOC1(setword,workspace,workspace_sz,5*m,"malloc"); for (v = 0; v < n; ++v)

{ gv = GRAPHROW(g,v,m);

EMPTYSET(gv,m);

ADDELEMENT(gv,(v+n-1)%n);

ADDELEMENT(gv,(v+1)%n);

} printf("Automorphisms of C[%d]:\n",n); nauty(g,lab,ptn,NULL,orbits,&options,&stats, workspace,5*m,m,n,NULL);

/* Get a pointer to the structure in which the group information has been stored.

If you use TRUE as an argument, the structure will be "cut loose" so that it won’t be used again the next time nauty() is called.

Otherwise, as here, the same structure is used repeatedly. */ group = groupptr(FALSE);

/* Expand the group structure to include a full set of coset representatives at every level.

This step is necessary if allgroup() is to be called. */ makecosetreps(group);

/* Call the procedure writeautom() for every element of the group.

The first call is always for the identity. */ allgroup(group,writeautom);

} else break;

} exit(0);

46

18.4

nautyex4.c : Sparse form with dynamic allocation

/* This program prints generators for the automorphism group of an n-vertex polygon, where n is a number supplied by the user.

It needs to be linked with nauty.c, nautil.c and nausparse.c.

This version uses sparse form with dynamic allocation.

*/

#include "nausparse.h" /* which includes nauty.h */ int main(int argc, char *argv[])

{

DYNALLSTAT(int,lab,lab_sz);

DYNALLSTAT(int,ptn,ptn_sz);

DYNALLSTAT(int,orbits,orbits_sz);

DYNALLSTAT(setword,workspace,workspace_sz); static DEFAULTOPTIONS_SPARSEGRAPH(options); statsblk stats; sparsegraph sg; /* Declare sparse graph structure */ int n,m,i; options.writeautoms = TRUE;

/* Initialise sparse graph structure. */

SG_INIT(sg); while (1)

{ printf("\nenter n : "); if (scanf("%d",&n) == 1 && n > 0)

{ m = (n + WORDSIZE - 1) / WORDSIZE; nauty_check(WORDSIZE,m,n,NAUTYVERSIONID);

DYNALLOC1(int,lab,lab_sz,n,"malloc");

DYNALLOC1(int,ptn,ptn_sz,n,"malloc");

DYNALLOC1(int,orbits,orbits_sz,n,"malloc");

DYNALLOC1(setword,workspace,workspace_sz,2*m,"malloc");

/* SG_ALLOC makes sure that the v,d,e fields of a sparse graph

* structure point to arrays that are large enough.

This only

* works if the structure has been initialised.

*/

SG_ALLOC(sg,n,2*n,"malloc");

47

} sg.nv = n; sg.nde = 2*n;

/* Number of vertices */

/* Number of directed edges */ for (i = 0; i < n; ++i)

{ sg.v[i] = 2*i; sg.d[i] = 2; sg.e[2*i] = (i+n-1)%n; sg.e[2*i+1] = (i+n+1)%n;

}

/* edge i->i-1 */

/* edge i->i+1 */ printf("Generators for Aut(C[%d]):\n",n); nauty((graph*)&sg,lab,ptn,NULL,orbits,&options,&stats, workspace,2*m,m,n,NULL); printf("Automorphism group size = %.0f",stats.grpsize1+0.1);

} else break;

} exit(0);

48

18.5

nautyex5.c : Determining an isomorphism, sparse form

4

0

9 1

5

2

6

8

2

3 7

0

7

1

9

6

5

8

Figure 8: Example of isomorphic graphs.

4

3

/* This program demonstrates how an isomorphism is found between graphs of the form in the figure above, for general size.

It needs to be linked with nauty.c, nautil.c and nausparse.c.

This version uses sparse form with dynamic allocation.

*/

#include "nausparse.h" /* which includes nauty.h */ int main(int argc, char *argv[])

{

DYNALLSTAT(int,lab1,lab1_sz);

DYNALLSTAT(int,lab2,lab2_sz);

DYNALLSTAT(int,ptn,ptn_sz);

DYNALLSTAT(int,orbits,orbits_sz);

DYNALLSTAT(setword,workspace,workspace_sz);

DYNALLSTAT(int,map,map_sz); static DEFAULTOPTIONS_SPARSEGRAPH(options); statsblk stats; sparsegraph sg1,sg2,cg1,cg2; /* Declare sparse graph structures */ int n,m,i;

/* Select option for canonical labelling */ options.getcanon = TRUE;

/* Initialise sparse graph structure. */

SG_INIT(sg1); SG_INIT(sg2);

SG_INIT(cg1); SG_INIT(cg2);

49

while (1)

{ printf("\nenter n : "); if (scanf("%d",&n) == 1 && n > 0)

{ if (n%2 != 0)

{ fprintf(stderr,"Sorry, n must be even\n"); continue;

} m = (n + WORDSIZE - 1) / WORDSIZE; nauty_check(WORDSIZE,m,n,NAUTYVERSIONID);

DYNALLOC1(int,lab1,lab1_sz,n,"malloc");

DYNALLOC1(int,lab2,lab2_sz,n,"malloc");

DYNALLOC1(int,ptn,ptn_sz,n,"malloc");

DYNALLOC1(int,orbits,orbits_sz,n,"malloc");

DYNALLOC1(setword,workspace,workspace_sz,2*m,"malloc");

DYNALLOC1(int,map,map_sz,n,"malloc");

/* Now make the first graph */

SG_ALLOC(sg1,n,3*n,"malloc"); sg1.nv = n; /* Number of vertices */ sg1.nde = 3*n; /* Number of directed edges */ for (i = 0; i < n; ++i)

{ sg1.v[i] = 3*i; sg1.d[i] = 3;

} for (i = 0; i < n; i += 2) sg1.e[sg1.v[i]] = i+1; for (i = 1; i < n; i += 2) sg1.e[sg1.v[i]] = i-1; for (i = 0; i < n; ++i)

{ sg1.e[sg1.v[i]+1] = i+2; sg1.e[sg1.v[i]+2] = i-2;

} sg1.e[sg1.v[0]+2] = n-1; sg1.e[sg1.v[1]+2] = n-2; sg1.e[sg1.v[n-2]+1] = 1; sg1.e[sg1.v[n-1]+1] = 0;

/* Now make the second graph */

50

SG_ALLOC(sg2,n,3*n,"malloc"); sg2.nv = n; /* Number of vertices */ sg2.nde = 3*n; /* Number of directed edges */ for (i = 0; i < n; ++i)

{ sg2.v[i] = 3*i; sg2.d[i] = 3;

} for (i = 0; i < n; ++i)

{ sg2.v[i] = 3*i; sg2.d[i] = 3; sg2.e[sg2.v[i]] = (i+1) % n; sg2.e[sg2.v[i]+1] = (i+n-1) % n; sg2.e[sg2.v[i]+2] = (i+n/2) % n;

}

/* Label sg1, result in cg1 and labelling in lab1; similarly sg1.

It is not necessary to pre-allocate space in cg1 and cg2, but they have to be initialised as we did above.

*/ nauty((graph*)&sg1,lab1,ptn,NULL,orbits,&options,&stats, workspace,2*m,m,n,(graph*)&cg1); nauty((graph*)&sg2,lab2,ptn,NULL,orbits,&options,&stats, workspace,2*m,m,n,(graph*)&cg2);

/* Compare canonically labelled graphs */ if (aresame_sg(&cg1,&cg2))

{ printf("Isomorphic.\n"); if (n <= 1000)

{

/* Write the isomorphism.

For each i, vertex lab1[i] of sg1 maps onto vertex lab2[i] of sg2.

We compute the map in order of labelling because it looks better. */ for (i = 0; i < n; ++i) map[lab1[i]] = lab2[i]; for (i = 0; i < n; ++i)

{ printf(" %d-%d",i,map[i]);

} printf("\n");

} else

}

51

} printf("Not isomorphic.\n");

} else break;

} exit(0);

52

19 Graph formats used by gtools programs

This is the file formats.txt.

Description of graph6 and sparse6 encodings

-------------------------------------------

Brendan McKay, [email protected]

Updated May 2005.

General principles:

All numbers in this description are in decimal unless obviously in binary.

Apart from the header, there is one object per line.

Apart from the header and the end-of-line characters, all bytes have a value in the range 63-126 (which are all printable ASCII characters).

A file of objects is a text file, so whatever end-of-line convention is locally used is fine).

Bit vectors:

A bit vector x of length k can be represented as follows.

Example: 1000101100011100

(1) Pad on the right with 0 to make the length a multiple of 6.

Example: 100010110001110000

(2) Split into groups of 6 bits each.

Example: 100010 110001 110000

(3) Add 63 to each group, considering them as bigendian binary numbers.

Example: 97 112 111

These values are then stored one per byte.

So, the number of bytes is ceiling(k/6).

Let R(x) denote this representation of x as a string of bytes.

Small nonnegative integers:

Let n be an integer in the range 0-68719476735 (2^36-1).

If 0 <= n <= 62, define N(n) to be the single byte n+63.

If 63 <= n <= 258047, define N(n) to be the four bytes

126 R(x), where x is the bigendian 18-bit binary form of n.

53

If 258048 <= n <= 68719476735, define N(n) to be the eight bytes

126 126 R(x), where x is the bigendian 36-bit binary form of n.

Examples: N(30) = 93

N(12345) = N(000011 000000 111001) = 126 69 63 120

N(460175067) = N(000000 011011 011011 011011 011011 011011)

= 126 126 63 90 90 90 90 90

Description of graph6 format.

----------------------------

Data type: simple undirected graphs of order 0 to 68719476735.

Optional Header:

>>graph6<< (without end of line!)

File name extension:

.g6

One graph:

Suppose G has n vertices.

Write the upper triangle of the adjacency matrix of G as a bit vector x of length n(n-1)/2, using the ordering

(0,1),(0,2),(1,2),(0,3),(1,3),(2,3),...,(n-1,n).

Then the graph is represented as N(n) R(x).

Example:

Suppose n=5 and G has edges 0-2, 0-4, 1-3 and 3-4.

x = 0 10 010 1001

Then N(n) = 68 and R(x) = R(010010 100100) = 81 99.

So, the graph is 68 81 99.

Description of sparse6 format.

------------------------------

Data type:

Undirected graphs of order 0 to 68719476735.

Loops and multiple edges are permitted.

Optional Header:

>>sparse6<< (without end of line!)

File name extension:

54

.s6

General structure:

Each graph occupies one text line.

Except for end-of-line characters, each byte has the form 63+x, where 0 <= x <= 63.

The byte encodes the six bits of x.

The encoded graph consists of:

(1) The character ’:’.

(This is present to distinguish the code from graph6 format.)

(2) The number of vertices.

(3) A list of edges.

(4) end-of-line

Loops and multiple edges are supported, but not directed edges.

Number of vertices n:

1, 4, or 8 bytes N(n) as above.

This is the same as graph6 format.

List of edges:

Let k be the number of bits needed to represent n-1 in binary.

The remaining bytes encode a sequence b[0] x[0] b[1] x[1] b[2] x[2] ... b[m] x[m]

Each b[i] occupies 1 bit, and each x[i] occupies k bits.

Pack them together in bigendian order, and pad up to a multiple of 6 as follows:

1. If (n,k) = (2,1), (4,2), (8,4) or (16,5), and vertex n-2 has an edge but n-1 doesn’t have an edge, and there are k+1 or more bits to pad, then pad with one

0-bit and enough 1-bits to complete the multiple of 6.

2. Otherwise, pad with enough 1-bits to complete the multiple of 6.

These rules are to match the gtools procedures, and to avoid the padding from looking like an extra loop in unusual cases.

Then represent this bit-stream 6 bits per byte as indicated above.

The vertices of the graph are 0..n-1.

The edges encoded by this sequence are determined thus: v = 0

55

for i from 0 to m do if b[i] = 1 then v = v+1 endif; if x[i] > v then v = x[i] else output {x[i],v} endif endfor

In decoding, an incomplete (b,x) pair at the end is discarded.

Example:

:[email protected]^

’:’ indicates sparse6 format.

Subtract 63 from the other bytes and write them in binary, six bits each.

000111 100010 000001 111001 011111

The first byte is not 63, so it is n.

n=7 n-1 needs 3 bits (k=3).

Write the other bits in groups of 1 and k:

1 000 1 000 0 001 1 110 0 101 1 111

This is the b/x sequence 1,0 1,0 0,1 1,6 0,5 1,7.

The 1,7 at the end is just padding.

The remaining parts give the edges 0-1 0-2 1-2 5-6.

For a description of the planarcode and edgecode formats, see the plantri documentation at http://cs.anu.edu.au/

∼ bdm/plantri

.

56

20 Help texts for gtools programs

===== addedgeg ======================================================

Usage: addedgeg [-lq] [-D#] [-btfF] [infile [outfile]]

For each edge e, output G-e

The output file has a header if and only if the input file does.

-l Canonically label outputs

-D# Specify an upper bound on the maximum degree of the output

-b Output is bipartite if input is

-t Output has no 3-cycles if input doesn’t

-f Output has no 4-cycles if input doesn’t

-F Output has no 5-cycles if input doesn’t

-btfF can be used in arbitrary combinations

-q Suppress auxiliary information

===== amtog ======================================================

Usage: amtog [-n#sghq] [infile [outfile]]

Read graphs in matrix format.

-n# Set the initial graph order to # (no default).

This can be overridden in the input.

-g

-s

-h

-q

Write the output in graph6 format (default).

Write the output in sparse6 format.

Write a header (according to -g or -s).

Suppress auxiliary information.

Input consists of a sequence of commands restricted to:

T q

M t n=# set number of vertices (no default)

The = is optional.

m Matrix to follow (01 any spacing or no spacing)

An ’m’ is also assumed if 0 or 1 is encountered.

Complement of matrix to follow (as m)

Upper triangle of matrix to follow, row by row excluding the diagonal. (01 in any or no spacing)

Complement of upper trangle to follow (as t) exit (optional)

57

===== biplabg ======================================================

Usage: biplabg [-q] [infile [outfile]]

Label bipartite graphs so that the colour classes are contiguous.

The first vertex of each component is assigned the first colour.

Vertices in each colour class have the same relative order as before.

Non-bipartite graphs are rejected.

The output file has a header if and only if the input file does.

-q Suppress auxiliary information.

===== catg ======================================================

Usage: catg [-xv] [infile]...

Copy files to stdout with all but the first header removed.

-x Don’t write a header.

In the absence of -x, a header is written if there is one in the first input file.

-v Summarize to stderr.

===== complg ======================================================

Usage: complg [-lrq] [infile [outfile]]

Take the complements of a file of graphs.

The output file has a header if and only if the input file does.

-r Only complement if the complement has fewer edges.

-l Canonically label outputs.

-q Suppress auxiliary information.

===== copyg ======================================================

Usage: copyg [-gsfp#:#qhx] [infile [outfile]]

Copy a file of graphs with possible format conversion.

-g Use graph6 format for output

-s Use sparse6 format for output

In the absence of -g and -s, the format depends on the header or, if none, the first input line.

58

-p# -p#:#

Specify range of input lines (first is 1)

-f With -p, assume input lines of fixed length

(ignored if header or first line has sparse6 format).

-h Write a header.

-x Don’t write a header.

In the absence of -h and -x, a header is written if there is one in the input.

-q Suppress auxiliary output.

===== countg ======================================================

Usage: [pickg|countg] [-fp#:#q -V] [--keys] [-constraints -v] [ifile [ofile]] countg : Count graphs according to their properties.

pickg : Select graphs according to their properties.

ifile, ofile : Input and output files.

’-’ and missing names imply stdin and stdout.

Miscellaneous switches:

-p# -p#:# Specify range of input lines (first is 1)

-f With -p, assume input lines of fixed length

(only used with a file in graph6 format)

-v

-V

-q

Negate all constraints

List properties of every input matching constraints.

Suppress informative output.

Constraints:

Numerical constraints (shown here with following #) can take a single integer value, or a range like #:#, #:, or :#.

Each can also be preceded by ’~’, which negates it.

(For example,

-~D2:4 will match any maximum degree which is _not_ 2, 3, or 4.)

Constraints are applied to all input graphs, and only those which match all constraints are counted or selected.

-n# number of vertices

-d# minimum degree

-r regular

-z# radius

-g# girth (0=acyclic)

-T# number of triangles

-e#

-D#

-b

-Z#

-Y# number of edges maximum degree bipartite diameter total number of cycles

-E Eulerian (all degrees are even, connectivity not required)

-a# group size -o# orbits -F# fixed points -t vertex-transitive

-c# connectivity (only implemented for 0,1,2).

59

-i# min common nbrs of adjacent vertices; -I# maximum

-j# min common nbrs of non-adjacent vertices; -J# maximum

Sort keys:

Counts are made for all graphs passing the constraints.

Counts are given separately for each combination of values occuring for the properties listed as sort keys.

A sort key is introduced by

’--’ and uses one of the letters known as constraints.

These can be combined: --n --e --r is the same as --ne --r and --ner.

The order of sort keys is significant.

===== deledgeg ======================================================

Usage: deledgeg [-lq] [-d#] [infile [outfile]]

For each edge e, output G-e

The output file has a header if and only if the input file does.

-l Canonically label outputs

-d# Specify a lower bound on the minimum degree of the output

-q Suppress auxiliary information

===== directg ======================================================

Usage: directg [-q] [-u|-T|-G] [-V] [-o] [-f#] [-e#|-e#:#] [infile [outfile]]

Read undirected graphs and orient their edges in all possible ways.

Edges can be oriented in either or both directions (3 possibilities).

Isomorphic directed graphs derived from the same input are suppressed.

If the input graphs are non-isomorphic then the output graphs are also.

-e# | -e#:# specify a value or range of the total number of arcs

-o orient each edge in only one direction, never both

-f# Use only the subgroup that fixes the first # vertices setwise

-T use a simple text output format (nv ne edges) instead of digraph6

-G like -T but includes group size as third item (if less than 10^10)

The group size does not include exchange of isolated vertices.

-V only output graphs with nontrivial groups (including exchange of isolated vertices).

The -f option is respected.

-u no output, just count them

-q suppress auxiliary information

===== dretog ======================================================

Usage: dretog [-n#o#sghq] [infile [outfile]]

60

Read graphs in dreadnaut format.

-o# Label vertices starting at # (default 0).

This can be overridden in the input.

-n# Set the initial graph order to # (no default).

-g

-s

-h

This can be overridden in the input.

Use graph6 format (default).

Use sparse6 format.

Write a header (according to -g or -s).

Input consists of a sequence of dreadnaut commands restricted to: n=# set number of vertices (no default)

The = is optional.

$=# set label of first vertex (default 0)

$$

The = is optional.

return origin to initial value (see -o#)

".." and !..\n comments to ignore g q specify graph to follow (as dreadnaut format)

Can be omitted if first character of graph is a digit or ’;’.

exit (optional)

===== genbg ======================================================

Usage: genbg [-c -ugsn -vq -lzF] [-Z#] [-d#|-d#:#] [-D#|-D#:#] n1 n2

[mine[:maxe]] [res/mod] [file]

Find all bicoloured graphs of a specified class.

n1 : the number of vertices in the first class n2 : the number of vertices in the second class mine:maxe : a range for the number of edges

#:0 means ’# or more’ except in the case 0:0 res/mod : only generate subset res out of subsets 0..mod-1 file : the name of the output file (defailt stdout)

-c : only write connected graphs

-z

-F

: all the vertices in the second class must have different neighbourhoods

: the vertices in the second class must have at least two

-L neighbours of degree at least 2

: there is no vertex in the first class whose removal leaves the vertices in the second class unreachable from each other

-Z# : two vertices in the second class may have at most # common nbrs

-D# : specify an upper bound for the maximum degree.

Example: -D6.

You can also give separate maxima for the two parts, for example: -D5:6

-d# : specify a lower bound for the minimum degree.

Again, you can specify it separately for the two parts: -d1:2

61

-n

-g

-s

-a

-u

-v

-l

: use nauty format for output

: use graph6 format for output (default)

: use sparse6 format for output

: use Greechie diagram format for output

: do not output any graphs, just generate and count them

: display counts by number of edges to stderr

: canonically label output graphs (using the 2-part colouring)

-q : suppress auxiliary output

See program text for much more information.

===== geng ======================================================

Usage: geng [-cCmtfbd#D#] [-uygsnh] [-lvq]

[-x#X#] n [mine[:maxe]] [res/mod] [file]

Generate all graphs of a specified class.

n : the number of vertices (1..32) mine:maxe : a range for the number of edges

#:0 means ’# or more’ except in the case 0:0 res/mod : only generate subset res out of subsets 0..mod-1

-u

-g

-s

-y

-h

-c

-C

-t

-f

-b

: only write connected graphs

: only write biconnected graphs

: only generate triangle-free graphs

: only generate 4-cycle-free graphs

: only generate bipartite graphs

(-t, -f and -b can be used in any combination)

-m : save memory at the expense of time (only makes a difference in the absence of -b, -t, -f and n <= 28).

-d# : a lower bound for the minimum degree

-D# : a upper bound for the maximum degree

-v : display counts by number of edges

-l : canonically label output graphs

: do not output any graphs, just generate and count them

: use graph6 output (default)

: use sparse6 output

: use the obsolete y-format instead of graph6 format

: for graph6 or sparse6 format, write a header too

-q : suppress auxiliary output (except from -v)

See program text for much more information.

62

===== genrang ======================================================

Usage: genrang [-P#|-P#/#|-e#|-r#|-R#] [-l#] [-m#] [-a] [-s|-g] [-S#] [-q] n num [outfile]

Generate random graphs.

n : number of vertices num : number of graphs

-s : Write in sparse6 format (default)

-g : Write in graph6 format

-P#/# : Give edge probability; -P# means -P1/#.

-e# : Give the number of edges

-r# : Make regular of specified degree

-R# : Make regular of specified degree but output as vertex count, edgecount, then list of edges

-l# : Maximum loop multiplicity (default 0)

-m# : Maximum multiplicity of non-loop edge (default and minimum 1)

-l and -m are only permitted with -R and -r without -g

-a : Make invariant under a random permutation

-S# : Specify random generator seed (default nondeterministic)

-q : suppress auxiliary output

Incompatible: -P,-e,-r,-R; -s,-g,-R; -R,-a; -s,-g & -l,-m.

===== gentourng ======================================================

Usage: gentourng [-cd#D#] [-ugs] [-lq] n [res/mod] [file]

Generate all tournaments of a specified class.

n : the number of vertices (1..32) res/mod : only generate subset res out of subsets 0..mod-1

-c : only write strongly-connected tournaments

-d# : a lower bound for the minimum out-degree

-D# : a upper bound for the maximum out-degree

-l : canonically label output graphs

-u : do not output any graphs, just generate and count them

-g

-s

: use graph6 output (lower triangle)

: use sparse6 output (lower triangle)

Default output is upper triangle row-by-row in ascii

-q : suppress auxiliary output

63

See program text for much more information.

===== labelg ======================================================

Usage: labelg [-qsg] [-fxxx] [-S] [-i# -I#:# -K#] [infile [outfile]]

Canonically label a file of graphs.

-s force output to sparse6 format

-g force output to graph6 format

If neither -s or -g are given, the output format is determined by the header or, if there is none, by the format of the first input graph. Also see -S.

-S Use sparse representation internally.

Note that this changes the canonical labelling.

Only sparse6 output is supported in this case.

Multiple edges are not supported.

One loop per vertex is ok.

The output file will have a header if and only if the input file does.

-fxxx Specify a partition of the point set.

xxx is any string of ASCII characters except nul.

This string is considered extended to infinity on the right with the character ’z’.

One character is associated with each point, in the order given.

The labelling used obeys these rules:

(1) the new order of the points is such that the associated characters are in ASCII ascending order

(2) if two graphs are labelled using the same string xxx, the output graphs are identical iff there is an associated-character-preserving isomorphism between them.

No option can be concatenated to the right of -f.

-i# select an invariant (1 = twopaths, 2 = adjtriang(K), 3 = triples,

4 = quadruples, 5 = celltrips, 6 = cellquads, 7 = cellquins,

8 = distances(K), 9 = indsets(K), 10 = cliques(K), 11 = cellcliq(K),

12 = cellind(K), 13 = adjacencies, 14 = cellfano, 15 = cellfano2)

-I#:# select mininvarlevel and maxinvarlevel (default 1:1)

-K# select invararg (default 3)

-q suppress auxiliary information

===== listg ======================================================

Usage: listg [-fp#:#l#o#Ftq] [-a|-A|-c|-d|-e|-M|-s] [infile [outfile]]

Write graphs in human-readable format.

-f : assume inputs have same size (only used from a file

64

and only if -p is given)

-p#, -p#:#, -p#-# : only display one graph or a sequence of graphs.

The first graph is number 1.

A second number which is empty or zero means infinity.

-a : write as adjacency matrix, not as list of adjacencies

-A : same as -a with a space between entries

-l# : specify screen width limit (default 78, 0 means no limit)

This is not currently implemented with -a or -A.

-o# : specify number of first vertex (default is 0).

-d : write output to satisfy dreadnaut

-c : write ascii form with minimal line-breaks

-e : write a list of edges, preceded by the order and the number of edges

-M : write in Magma format

-W : write matrix in Maple format

-t : write upper triangle only (affects -a, -A, -d and default)

-s : write only the numbers of vertices and edges

-F : write a form-feed after each graph except the last

-q : suppress auxiliary output

-a, -A, -c, -d, -M, -W and -e are incompatible.

===== multig ======================================================

Usage: multig [-q] [-u|-T|-G|-A|-B] [-e#|-e#:#] [-m#] [-f#] [infile [outfile]]

Read undirected loop-free graphs and replace their edges with multiple edges in all possible ways (multiplicity at least 1).

Isomorphic multigraphs derived from the same input are suppressed.

If the input graphs are non-isomorphic then the output graphs are also.

-e# | -e#:# specify a value or range of the total number of edges counting multiplicities

-m# maximum edge multiplicity (minimum is 1)

Either -e or -m with a finite maximum must be given

-f# Use the group that fixes the first # vertices setwise

-T use a simple text output format (nv ne {v1 v2 mult})

-G like -T but includes group size as third item (if less than 10^10)

The group size does not include exchange of isolated vertices.

-A write as the upper triangle of an adjacency matrix, row by row, including the diagonal, and preceded by the number of vertices

-B write as an integer matrix preceded by the number of rows and number of columns, where -f determines the number of rows

65

-u no output, just count them

-q suppress auxiliary information

===== newedgeg ======================================================

Usage: newedgeg [-lq] [infile [outfile]]

For each pair of non-adjacent edges, output the graph obtained by subdividing the edges and joining the new vertices.

The output file has a header if and only if the input file does.

-l Canonically label outputs

-q Suppress auxiliary information

===== NRswitchg ======================================================

Usage: NRswitchg [-lq] [infile [outfile]]

For each v, complement the edges from N(v) to V(G)-N(v)-v.

The output file has a header if and only if the input file does.

-l Canonically label outputs.

-q Suppress auxiliary information.

===== pickg ======================================================

Usage: [pickg|countg] [-fp#:#q -V] [--keys] [-constraints -v] [ifile [ofile]] countg : Count graphs according to their properties.

pickg : Select graphs according to their properties.

ifile, ofile : Input and output files.

’-’ and missing names imply stdin and stdout.

Miscellaneous switches:

-p# -p#:# Specify range of input lines (first is 1)

-f

-v

With -p, assume input lines of fixed length

(only used with a file in graph6 format)

Negate all constraints

-V

-q

List properties of every input matching constraints.

Suppress informative output.

Constraints:

Numerical constraints (shown here with following #) can take a single integer value, or a range like #:#, #:, or :#.

Each

66

can also be preceded by ’~’, which negates it.

(For example,

-~D2:4 will match any maximum degree which is _not_ 2, 3, or 4.)

Constraints are applied to all input graphs, and only those which match all constraints are counted or selected.

-n# number of vertices

-d# minimum degree

-r regular

-e# number of edges

-D# maximum degree

-b bipartite

-z# radius

-g# girth (0=acyclic)

-Z# diameter

-Y# total number of cycles

-T# number of triangles

-E Eulerian (all degrees are even, connectivity not required)

-a# group size -o# orbits -F# fixed points -t vertex-transitive

-c# connectivity (only implemented for 0,1,2).

-i# min common nbrs of adjacent vertices; -I# maximum

-j# min common nbrs of non-adjacent vertices; -J# maximum

Sort keys:

Counts are made for all graphs passing the constraints.

Counts are given separately for each combination of values occuring for the properties listed as sort keys.

A sort key is introduced by

’--’ and uses one of the letters known as constraints.

These can be combined: --n --e --r is the same as --ne --r and --ner.

The order of sort keys is significant.

===== planarg ======================================================

Usage: planarg [-v] [-nVq] [-p|-u] [infile [outfile]]

For each input, write to output if planar.

The output file has a header if and only if the input file does.

-v Write non-planar graphs instead of planar graphs

-V Write report on every input

-u Don’t write anything, just count

-p Write in planar_code if planar (without -p, same format as input)

-k Follow each non-planar output with an obstruction in sparse6 format (implies -v, incompatible with -p)

-n Suppress checking of the result

-q Suppress auxiliary information

This program permits multiple edges and loops

67

===== shortg ======================================================

Remove isomorphs from a file of graphs.

If outfile is omitted, it is taken to be the same as infile

If both infile and outfile are omitted, input will be taken from stdin and written to stdout

The output file has a header if and only if the input file does.

-s force output to sparse6 format

-g force output to graph6 format

If neither -s or -g are given, the output format is determined by the header or, if there is none, by the format of the first input graph.

-k output graphs have the same labelling and format as the inputs.

Otherwise, output graphs have canonical labelling.

-s and -g are ineffective if -k is given.

If none of -sgk are given, the output format is determined by the header or, if there is none, by the format of the first input graph.

-v write to stderr a list of which input graphs correspond to which output graphs. The input and output graphs are both numbered beginning at 1.

A line like

23 : 30 154 78 means that inputs 30, 154 and 78 were isomorphic, and produced output 23.

-d include in the output only those inputs which are isomorphic to another input.

If -k is specified, all such inputs are included in their original labelling.

Without -k, only one member of each nontrivial isomorphism class is written, with canonical labelling.

-fxxx Specify a partition of the point set.

xxx is any string of ASCII characters except nul.

This string is considered extended to infinity on the right with the character ’z’.

One character is associated with each point, in the order given.

The labelling used obeys these rules:

(1) the new order of the points is such that the associated characters are in ASCII ascending order

(2) if two graphs are labelled using the same string xxx, the output graphs are identical iff there is an associated-character-preserving isomorphism between them.

-i# select an invariant (1 = twopaths, 2 = adjtriang(K), 3 = triples,

4 = quadruples, 5 = celltrips, 6 = cellquads, 7 = cellquins,

68

8 = distances(K), 9 = indsets(K), 10 = cliques(K), 11 = cellcliq(K),

12 = cellind(K), 13 = adjacencies, 14 = cellfano, 15 = cellfano2)

-I#:# select mininvarlevel and maxinvarlevel (default 1:1)

-K# select invararg (default 3)

-u Write no output, just report how many graphs it would have output.

In this case, outfile is not permitted.

-Tdir Specify that directory "dir" will be used for temporary disk space by the sort subprocess.

The default is usually /tmp.

-q Suppress auxiliary output

69

References

[1] J. M. Boyer and W. J. Myrvold, On the cutting edge: simplified O(n) planarity by edge addition, J. Graph Alg. Appl., 8 (2004) 241–273.

[2] P. T. Darga, M. H. Liffiton, K. A. Sakallah, and I. L. Markov, Exploiting Structure in

Symmetry Generation for CNF, Proceedings of the 41st Design Automation Conference, 2004, 530–534. Source code at http://vlsicad.eecs.umich.edu/BK/SAUCY

.

[3] B. D. McKay, Hadamard equivalence via graph isomorphism, Discrete Math., 27

(1979) 213–214.

[4] A. Kirk, Efficiency considerations in the canonical labelling of graphs, Technical report TR-CS-85-05, Computer Science Department, Australian National University

(1985).

[5] B. D. McKay, A. Meynert and W. Myrvold, Small Latin squares, quasigroups and loops, J. Combin. Designs, to appear.

[6] T. Miyazaki, The complexity of McKay’s canonical labelling algorithm, in Groups and Computation, II, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 28, Amer.

Math. Soc. (1997) 239–256.

[7] K. E. Malysiak, Graph Isomorphism, Canonical Labelling and Invariants, Honours

Thesis, Computer Science Department, Australian National University (1987).

[8] R. Mathon, Sample graphs for isomorphism testing, Congressus Numerantium, 21

(1978) 499–517.

[9] B. D. McKay, Practical graph isomorphism, Congressus Numerantium, 30 (1981)

45–87. Available at http://cs.anu.edu.au/

∼ bdm/nauty/PGI

.

70

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