# Digraphs

Graphs, digraphs, and multidigraphs in GAP

Version 1.6.1
Released 2022-12-06

This project is maintained by James Mitchell, Wilf A. Wilson

# CHANGELOG – Digraphs package for GAP

Copyright © 2014-22 by Jan De Beule, Julius Jonušas, James D. Mitchell, Wilf A. Wilson, Michael Young et al.

Licensing information can be found in the LICENSE file.

## Version 1.5.2 (released 30/03/2022)

This is a very minor release containing technical changes for maintaining compatibility with other GAP packages.

## Version 1.5.1 (released 29/03/2022)

This minor release contains several bugfixes and technical changes. This includes:

• Bugfix: vertex labels are no longer wrongly retained when using DigraphEdgeUnion. This was reported by Wilf A. Wilson in Issue #496 and fixed by Joseph Edwards in PR #507.
• Bugfix: a segfault could be caused by calling OutNeighbours with an inappropriate argument. This was reported by Wilf A. Wilson in Issue #518 and fixed by James D. Mitchell in PR #519.
• Wilf A. Wilson improved the performance of DigraphAddEdge for digraphs without edge labels in PR #509.
• Max Horn changed the declaration of the variable Vertices to improve compatibility with Grape in PR #530.

## Version 1.5.0 (released 27/10/2021)

This is a fairly major release of the Digraphs package, containing some bugfixes and several new features.

In this version, we welcome Finn Buck, Tom Conti-Leslie, Ewan Gilligan, Lea Racine, and Ben Spiers as contributors to the package.

### Bugfixes

• The edge labels of Cayley digraphs could sometimes be incorrect (Fixed by Jan De Beule in PR #452)
• Typos in the documentation of IsDirectedTree and an error message for OnDigraphs were fixed (Wilf A. Wilson in PRs #480 and #498)

### A database of one-off named graphs and digraphs, and more families of standard examples

We especially wish to highlight the greatly expanded functionality for creating digraphs that are either famous one-off examples, or are part of a family of standard examples.

In particular, Marina Anagnostopoulou-Merkouri, Finn Buck, James D. Mitchell, Lea Racine, and Ben Spiers implemented functions to construct many more families of standard examples (currently documented in Section 3.5), which were added in in PRs #408, #409, #411, #415, #416, #417, #423, #424, #425, #445, #454, #456, and #490.

Furthermore, Marina Anagnostopoulou-Merkouri, Reinis Cirpons, Tom Conti-Leslie, Lea Racine, Maria Tsalakou, and Murray Whyte added a database of one-off named graphs and digraphs in PR #404. These digraphs can be constructed by calling Digraph with a string of appropriate name, e.g. Digraph("brinkmann"). The available names can be accessed with the ListNamedDigraphs function.

### Other new features

• Tarjan and Lengauer’s almost-linear time dominators algorithm was implemented, and is available via Dominators and DominatorTree (James D. Mitchell, Marina Anagnostopoulou-Merkouri, Samantha Harper, and Finn Buck, in PR #336)
• MaximalCommonSubdigraph and MinimalCommonSuperdigraph were introduced (Luke Elliot, PR #361)
• DigraphShortestPathSpanningTree was introduced (Jan De Beule and Wilf A. Wilson, in PR #363)
• Lawler and Byskov’s algorithms for chromatic number were implemented (Ewan Gilligan, PR #382)
• Cayley digraphs now have pre-set vertex and edge labels (Jan De Beule and Wilf A. Wilson, in PR #385)
• DigraphCycle was added as a synonym for CycleDigraph (Wilf A. Wilson, PR #441)
• Several new digraph product operations were introduced: StrongProduct, ConormalProduct, HomomorphicProduct, and LexicographicProduct (Finn Buck, PR #460)
• The operation IsDigraphPath was introduced (James D. Mitchell, PR #489)

## Version 1.4.1 (released 14/05/2021)

This minor release contains several bugfixes and technical changes, and improvements to the documentation. These include the following:

• OutNeighbours is now a global function rather than an attribute (James D. Mitchell, PR #413).
• The configuration options for Digraphs are now described in the manual (James D. Mitchell, PR #420).
• The included version of the Edge Addition Planarity Suite has been updated from version 3.0.0.5 to 3.0.1.0, and is now documented in the manual (James D. Mitchell, PRs #421, #422).
• SetDigraphVertexLabels now accepts an immutable list of labels (Murray Whyte, PR #427).
• A bug was fixed in DigraphCore that could lead to wrong results (Wilf A. Wilson, PR #437).
• The performance of ChromaticNumber was improved in some cases by the use of Brooks’ theorem (Wilf A. Wilson, PR #446).
• A bug, reported by Leonard Soicher, was fixed in DigraphMaximumMatching which affected digraphs with custom vertex labels (Wilf A. Wilson, PR #462).
• A bug was fixed that affected DigraphEdgeUnion, DigraphJoin, DigraphDisjointUnion, DigraphDirectProduct, and DigraphCartesianProduct when each was given a single list of digraphs as the argument. (Wilf A. Wilson, PR #468).

## Version 1.4.0 (released 27/01/2021)

In this release there are several new features and improvements.

The following improvements and bugfixes have been made:

• the clique finder was reimplemented in C by Julius Jonusas
• a critical bug in CayleyDigraph was reported and fixed by Jan De Beule
• a minor bug in ReadDigraphs was reported by Wilf A. Wilson and fixed by James D. Mitchell
• the performance of BlissCanonicalLabelling and BlissAutomorphismGroup was improved for multidigraphs by Marina Anagnostopoulou-Merkouri and Sam Harper.
• a bug in GeneratorsOfEndomorphismMonoid that caused GAP to crash when called with a multidigraph was reported by Wilf A. Wilson and fixed by James D. Mitchell
• Wilf A. Wilson made some improvements to the manual.
• the performance of DigraphCopy was improved by Marina Anagnostopoulou-Merkouri and Sam Harper.

The main new features are:

• the attribute DigraphNrLoops was introduced by Marina Anagnostopoulou-Merkouri and Sam Harper.
• the operations
• DotColoredDigraph
• DotVertexColoredDigraph
• DotEdgeColoredDigraph
• DotSymmetricColoredDigraph
• DotSymmetricVertexColoredDigraph
• DotSymmetricEdgeColoredDigraph were introduced by Marina Anagnostopoulou-Merkouri and Sam Harper.
• the operation VerticesReachableFrom was introduced by Marina Anagnostopoulou-Merkouri.
• the operation ModularProduct was introduced by Luke Elliott and
James D. Mitchell

## Version 1.3.1 (released 27/11/2020)

This is a minor release fixing some issues, some performance improvements, and removing some deprecated code. The changes in this release were made by Max Horn and Wilf A. Wilson.

## Version 1.3.0 (released 27/06/2020)

This is a minor release adding the new functionality DigraphMutabilityFilter, StrongOrientation, Bridges, and IsBridgeless James D. Mitchell.

## Version 1.2.1 (released 27/05/2020)

This is a minor release where some of the documentation has been fixed and the installation instructions have been improved James D. Mitchell, some changes were made for compatibility with future versions of GAP Max Horn and Wilf A. Wilson.

## Version 1.2.0 (released 27/03/2020)

This is a minor release adding some new features to Digraphs, principally functionality relating to computing matchings by Reinis Ciprons, and an implementation of Dijkstra’s algorithm for shortest paths by Markus Pfeiffer and Maria Tsalakou, and methods for producing a concise string representation of a digraph by Murray Whyte.

As of this version, Digraphs requires the datastructures package to be available, in version 0.2.5 or newer.

## Version 1.1.2 (released 16/03/2020)

This is a minor release adding the new configuration flag --without-intrinsics and checking that the compiler is in C99 mode by using AC_PROG_CC_C99 in configure.ac.

## Version 1.1.1 (released 29/01/2020)

This release fixes a bug in HomomorphismDigraphsFinder that was introduced in version 1.1.0. The bug was found and fixed by James D. Mitchell; see PR #290.

## Version 1.1.0 (released 25/01/2020)

This is a minor release that includes some new features and some performance improvements.

The following issues were resolved, pull requests merged, or new features added:

• Issue #40: If bliss is used to compute the automorphism group of a digraph, then the size of the automorphism group is returned from bliss to GAP, and the group object in GAP immediately knows its size. In particular, it is not necessary to recalculate this size. This was reported and fixed in PR #278 by Chris Jefferson.

• Issue #279: In the function HomomorphismDigraphsFinder, it is now possible to specify a subgroup of the automorphism group of the range digraph. This way, the automorphism group of the range digraph is not computed by HomomorphismDigraphsFinder. This can result in a performance improvement in some cases. This was reported and fixed in PR #285 by Finn Smith.

• Issue #284: The function HomomorphismDigraphsFinder sometimes did not return any homomorphisms when the source digraph had exactly one vertex. This was caused by the data structures used by HomomorphismDigraphsFinder not being correctly initialised in this case. This issue was reported by Finn Smith and fixed by James D. Mitchell in PR #286.

• In PR #283, Finn Smith added the new operation DigraphsRespectsColouring, which can be used to check whether a transformation or permutation between digraphs respects given colourings. New versions of IsDigraphHomomorphism, and friends, were added that accept colourings as arguments and which use DigraphsRespectsColouring.

• The version of bliss included in Digraphs was updated to allow all of its data structures to be modified in-place rather than allocated and deallocated repeatedly. The function HomomorphismDigraphsFinder was modified to make use of this new functionality in bliss, and subsequently the performance of HomomorphismDigraphsFinder has been improved (particularly in cases where many homomorphisms between distinct small digraphs are found). This was done by James D. Mitchell in PR #282.

• Some further minor performance improvements were made, and a compiler warning was fixed.

## Version 1.0.3 (released 29/11/2019)

This is a minor release that fixes some bugs related to mutability in DigraphDisjointUnion and ViewString.

These problems were reported and fixed by Wilf A. Wilson in Issue #276 and PR #277, respectively.

## Version 1.0.2 (released 28/11/2019)

This is a minor release that fixes several bugs:

## Version 1.0.1 (released 05/10/2019)

This is a minor release of the Digraphs package. The main change in this release is the reintroduction of the three-argument version of DigraphAddVertices, which accepts a digraph, a number of vertices to add, and a list of labels for the new vertices. The removal inadvertantly broke backwards compatbility with some third-party pre-existing code that relied on this functionality in the Digraphs package (see Issue #264).

The second argument of the three-argument version was redundant, and so a new two-argument version of DigraphAddVertices, which accepts a digraph and a list of new vertex labels, was introduced in v1.0.0. Unfortunately, the concurrent removal of the three-argument version of DigraphAddVertices was not advertised in the CHANGELOG. Although the three-argument version has been reintroduced, it will remain undocumented, since there is no good reason for any new code to use the three-argument version.

The author contact data on the title page of the manual was also updated.

The changes in this version were made by Wilf A. Wilson.

## Version 1.0.0 (released 03/10/2019)

This is a major release of the Digraphs package that introduces significant new functionality, some changes in behaviour, general improvements, and several small bugfixes. With this version, we welcome Reinis Cirpons as a contributor to the package.

### Changed functionality or names

• Perhaps the most immediately visible change is that the ViewString for immutable digraphs attempts to show more of the known information about the digraph. This will break tests that relied on the previous behaviour, that contained only the numbers of vertices and edges.
• The behaviour of QuotientDigraph has been changed so that it no longer returns digraphs with multiple edges.
• IsEulerianDigraph would previously return true for digraphs that are Eulerian when their isolated vertices were removed, which contradicted the documentation. IsEulerianDigraph now returns false for all digraphs that are not strongly connected.
• The type of all digraphs has been renamed from DigraphType to DigraphByOutNeighboursType.
• The synonym DigraphColoring (American-style spelling) for DigraphColouring was removed.

### Immutable and mutable digraphs

Previously, every digraph in the Digraphs package was an immutable, attribute-storing digraph. It is now possible to create mutable digraphs. Mutable digraphs are not attribute-storing, but they can be altered in place - by adding or removing vertices and edges - which, unlike with immutable digraphs, does not require a new copy of the digraph to be made. This can save time and memory.

This is particularly useful when one wants to create a digraph, alter the digraph in some way, and then perform some computations. One can now typically do this with fewer resources by creating a mutable digraph, modifying it in place, and then converting it into an immutable digraph (which can store attributes and properties), before finally performing the computations.

Every digraph now belongs to precisely one of the categories IsMutableDigraph or IsImmutableDigraph, according to its mutability. A mutable digraph can be converted in-place into an immutable digraph with MakeImmutable. The are various new and updated functions for creating mutable and immutable digraphs, and for making mutable or immutable copies.

Most digraph-creation functions in the package now accept an optional first argument, that can be either IsMutableDigraph or IsImmutableDigraph. Given one of these filters, the function will according create the digraph to be of the appropriate mutability. When this is option available, the default is always to create an immutable digraph.

On the whole, for a function in the package that takes a digraph as its argument and again returns a digraph, the function now returns a digraph of the same mutability as its result, and moreover, given a mutable argument, it converts the mutable digraph in-place into the result. However, please consult the document to learn the exact behaviour of any specific function.

Old attributes Foo in the package that take and return a single digraph have been converted into the operation Foo, with a corresponding new attribute, FooAttr. This means that the getter and setter functions, HasFoo and SetFoo, are renamed to HasFooAttr and SetFooAttr. See DigraphReverse for an example. For an immutable (and therefore attribute-storing) digraph, calling Foo calls FooAttr and returns an immutable digraph, which it stores, and so the effect is as before. For an mutable digraph, calling Foo modifies the digraph in-place, which remains mutable.

The majority of the changes in Digraphs relating to mutable and immutable digraphs were made by James D. Mitchell, Finn Smith, and Wilf A. Wilson, with some further contributions by Reinis Cirpons, Luke Elliott, and Murray Whyte.

### New and extended functions

The package now includes the following new functions:

• AsSemigroup can produce strong semilattices of groups (i.e. Clifford) from semilattice digraphs, groups, and homomorphisms. This functionality was added by Finn Smith in PR #161.
• AutomorphismGroup and BlissAutomorphismGroup can now take an optional third argument that specifies an edge colouring for the digraph. In this case, the functions return only automorphisms of the digraph that preserve the edge colouring (and the vertex colouring, if one is given). This brilliant new functionality was added by Finn Smith in PR #186.
• DegreeMatrix, LaplacianMatrix, and NrSpanningTrees were introduced by Reinis Cirpons in PR #224.
• DigraphCartesianProduct and DigraphDirectProduct, along with the companion functions DigraphCartesianProductProjections and DigraphDirectProductProjections, were introduced by Reinis Cirpons in PR #228.
• DigraphMycielskian was added by Murray Whyte in PR #194.
• DigraphNrStronglyConnectedComponents was added by Murray Whyte in PR #180.
• DigraphOddGirth was added by Murray Whyte in PR #166
• DigraphCore and IsDigraphCore were added by Murray Whyte in PRs #221 and #217, respectively.
• DotHighlightedDigraph was added by Finn Smith in PR #169.
• IsCompleteMultipartiteDigraph was added by Wilf A. Wilson in PR #236.
• IsEquivalenceDigraph was added by Wilf A. Wilson in PR #234 as a synonym for IsReflexiveDigraph and IsSymmetricDigraph and IsTransitiveDigraph.
• IsVertexTransitive and IsEdgeTransitive were added by Graham Campbell in PR #165.
• PetersenGraph and GeneralisedPetersenGraph were added by Murray Whyte in PRs #181 and #204, respectively.
• RandomLattice was added by Reinis Cirpons in PR #175.

### New technical functionality

• The ability to compile (with the flag --with-external-bliss) and use the Digraphs package with the system version of bliss was added by Isuru Fernando in PR #225.
• The ability to compile (with the flag --with-external-planarity) and use the Digraphs package with the system version of the Edge Addition Planarity Suite was added by James D. Mitchell in PR #207.

## Version 0.15.4 (released 06/08/2019)

This is a minor release that fixes a few bugs.

In previous versions, the homomorphism-finding tools sometimes returned purported ‘monomoprhisms’ that were not injective. This problem was reported by Gordon Royle, see Issue #222, and fixed by James D. Mitchell in PR #223. In addition, Wilf A. Wilson fixed a bug in DigraphNrEdges. This function could previously lead to a crash when given a digraph whose OutNeighbours contained entries not in IsPlistRep.

## Version 0.15.3 (released 12/06/2019)

This is a minor release that fixes a typo in the documentation of JohnsonDigraph, and contains some minor tweaks for compatibility with future versions of GAP.

## Version 0.15.2 (released 17/04/2019)

This is a minor release that updates Digraphs for compatibility with the upcoming GAP 4.11, and resolves a bug in IsHamiltonianDigraph that could have lead to the boolean adjacency matrix of a digraph being accidentally modified; see Issue #191 and PR #192.

## Version 0.15.1 (released 26/03/2019)

This is a minor release of the Digraphs package, which improves the compatibility of Digraphs with cygwin. In particular, in the Windows installer of the next release of GAP, Digraphs should be included in a pre-compiled and working state. See Issue #177 and PR #178 for more details.

Digraphs now requires version 4.8.2 of the orb package, or newer.

## Version 0.15.0 (released 15/02/2019)

This release contains several substantial new features, and some changes to previous functionality.

The most significant change in behaviour is related to the Digraph6 format used in previous versions of the Digraphs package. This method of encoding directed graphs was developed independently from, but concurrently with, the Digraph6 format introduced by nauty; see Issue #158 for more information. The Digraphs package now uses the nauty format, although digraphs encoded using the old format can still be read in. This incompatibility was reported by Jukka Kohonen, and the changes were made by Michael Young in PR #162.

Other additions and changes are listed below:

• A copy of the Edge Addition Planarity Suite is now included in Digraphs, and so it is now possible to test digraphs for planarity, and to perform related computations. This was added by James D. Mitchell in PR #156. The new functionality can be accessed via:
• Is(Outer)PlanarDigraph,
• (Outer)PlanarEmbedding,
• Kuratowski(Outer)PlanarSubdigraph,
• SubdigraphHomeomorphicToK(23/4/33), and
• MaximalAntiSymmetricSubdigraph.
• The functionality and performance for computing homomorphisms of digraphs was significantly improved by James D. Mitchell in PR #160. This PR also introduced the operations EmbeddingsDigraphs and EmbeddingsDigraphsRepresentatives.
• The one-argument attribute DigraphColouring was renamed to DigraphGreedyColouring, and its performance was improved; it now uses the Welsh-Powell algorithm, which can be accessed directly via DigraphWelshPowellOrder. The behaviour of DigraphGreedyColouring can be modified by including an optional second argument; see the documentation for more information. This work was done by James D. Mitchell in PR #144.
• DigraphShortestPath was introduced by Murray Whyte in PR #148.
• IsAntiSymmetricDigraph (with a capital S) was added as a synonym for IsAntisymmetricDigraph.
• RandomDigraph now allows a float as its second argument; by James D. Mitchell in PR #159.
• The attribute CharacteristcPolynomial for a digraph was added by Luke Elliott in PR #164.
• The properties IsVertexTransitive and IsEdgeTransitive for digraphs were added by Graham Campbell in PR #165.

## Version 0.14.0 (released 23/11/2018)

This release contains bugfixes and a couple of new features.

• The operations AsSemigroup and AsMonoid for lattice and semilattice digraphs were added by Chris Russell in PR #136.
• The operation IsDigraphColouring was added by James D. Mitchell in PR #145.
• In previous versions of the package, the output of ArticulationPoints would sometimes contain repeated vertices (reported by Luke Elliott in Issue #140, and fixed by James D. Mitchell in PR #142).
• In previous versions of the package, an unexpected error was sometimes caused when removing an immutable set of vertices from a digraph (reported and fixed by James D. Mitchell in PR #146).
• The header file x86intrin.h was unnecessarily being included by the kernel module of Digraphs (reported by Wilf A. Wilson in Issue #147, and fixed by James D. Mitchell in PR #152).

Max Horn also contributed various compatibility and correctness changes to the kernel module of the package, including in PRs #149, #150, and #151.

Digraphs now requires version 4.8.1 of the orb package, or newer.

## Version 0.13.0 (released 19/09/2018)

This release of Digraphs contains some bugfixes, along with the following new features:

• The GraphViz engine used by Splash is now configurable, thanks to Markus Pfeiffer.
• The properties IsPartialOrderDigraph, IsPreorderDigraph, and IsQuasiorderDigraph were introduced by Chris Russell, along with the following functions for visualising these kinds of digraphs:
• DotPartialOrderDigraph
• DotPreorderDigraph
• DotQuasiorderDigraph
• The following functions for transformations and permutations were added by James D. Mitchell:
• IsDigraphHomomorphism
• IsDigraphEpimorphism
• IsDigraphMonomorphism
• IsDigraphEndomorphism
• IsDigraphEmbedding
• IsDigraphIsomorphism

Digraphs now requires version 4.9.0 of GAP, or newer.

## Version 0.12.2 (released 24/08/2018)

This is a minor release which contains some small adjustments to the build system of the package.

## Version 0.12.1 (released 26/04/2018)

This is a minor release, which contains several bugfixes. The following problems were resolved by James D. Mitchell:

• HomomorphismDigraphFinder sometimes failed to find a homomorphism when one existed [Issue #111, reported by Gordon Royle];
• the documentation for HomomorphismDigraphFinder was incomplete [Issue #112]; and
• a segmentation fault could be caused when using Digraphs with NautyTracesInterface, in certain cases [Issue #114].

## Version 0.12.0 (released 31/01/2018)

This release contains bugfixes and new features. In particular, it:

Digraphs now requires version 4.5.1 of the IO package.

## Version 0.11.0 (released 22/11/2017)

The principal change in Digraphs version 0.11.0 is the addition of support for computing automorphisms, canonical labellings, and isomorphisms of digraphs with nauty. This functionality requires the NautyTracesInterface package for GAP, version 0.2 or newer. However, this is not a required package, and the default engine remains bliss. It is possible to specify the engine that is used by Digraphs. These changes to Digraphs were made by James D. Mitchell].

In particular, version 0.11.0 includes the following changes:

• BlissAutomorphismGroup and NautyAutomorphismGroup are introduced.
• DigraphCanonicalLabelling is replaced by BlissCanonicalLabelling and NautyCanonicalLabelling.
• BlissCanonicalDigraph and NautyCanonicalDigraph are introduced [Chris Russell and James D. Mitchell].
• DigraphsUseNauty and DigraphsUseBliss are introduced.

The property IsHamiltonianDigraph and the attribute HamiltonianPath were added by Luke Elliott. Additionally, this release fixes several bugs, including one in DigraphSymmetricClosure and one in CompleteDigraph.

Digraphs now requires version 4.4.6 of the IO package.

## Version 0.10.1 (released 16/08/2017)

This is a minor release, which contains performance improvements, and fixes a bug in Digraph that could cause a segmentation fault.

## Version 0.10.0 (released 20/07/2017)

This release contains new features, bugfixes, and minor improvements to the documentation. There is a new method for ChromaticNumber, which has better performance than the previous method [Julius Jonusas and James D. Mitchell]. A bug in the code for calculating homomorphisms of digraphs, which could cause a crash, was resolved [James D. Mitchell].

### New Features in Version 0.10.0

• Vertex labelled digraphs can now be visualised in a way that displays vertex labels, by using the new operation DotVertexLabelledDigraph.
• The attribute CliqueNumber is introduced.
• The following new attributes for Cayley digraphs are introduced:
• GroupOfCayleyDigraph
• SemigroupOfCayleyDigraph
• GeneratorsOfCayleyDigraph

All of the new features were added by James D. Mitchell.

## Version 0.9.0 (released 12/07/2017)

This release introduces several new features.

### New Features in Version 0.9.0

The following attributes and properties were added by James D. Mitchell:

• ArticulationPoints (and its synonym CutVertices)
• IsBiconnectedDigraph
• IsCycleDigraph

The following operations related to matchings were added by Isabella Scott and Wilf A. Wilson:

• IsMatching
• IsPerfectMatching
• IsMaximalMatching

## Version 0.8.1 (released 18/05/2017)

This is a minor release, which updates the README file and updates the list of package authors and contributors.

## Version 0.8.0 (released 17/05/2017)

This release contains new features, several minor bugfixes, and minor improvements to the documentation of the package.

### New Features in Version 0.8.0

This release introduces the new operations DigraphClosure [Julius Jonusas] and BooleanAdjacencyMatrixMutableCopy [Wilf A. Wilson], along with the following properties and operations related to semilattices [Chris Russell]:

• IsPartialOrderDigraph
• IsMeetSemilatticeDigraph
• IsJoinSemilatticeDigraph
• IsLatticeDigraph
• PartialOrderDigraphMeetOfVertices
• PartialOrderDigraphJoinOfVertices

## Version 0.7.1 (released 22/03/2017)

This is a minor release, which fixes bugs in DigraphTopologicalSort and IsAntisymmetricDigraph that could trigger segmentation faults.

## Version 0.7.0 (released 14/03/2017)

This release introduces several new features, changes some existing functionality, and improves the documentation. The changes in this release were made by Wilf A. Wilson.

### New Features in Version 0.7.0

• This release contains a new technique for encoding a vertex-coloured multidigraph as a vertex-coloured (undirected) graph while preserving the automorphism group, in order to calculate the automorphism group and canonical labelling using bliss. This enables the following functionality:
• the operations AutomorphismGroup and DigraphCanonicalLabelling for a digraph and a vertex-colouring now accept a multidigraph as their first argument;
• the operations IsIsomorphicDigraph and IsomorphismDigraphs now accept multidigraphs, and they also accept vertex-colourings as optional arguments.
• This release add new functionality related to undirected spanning trees and undirected spanning forests:
• the property IsUndirectedForest is introduced;
• the attributes UndirectedSpanningTree and UndirectedSpanningForest are introduced; and
• the operations IsUndirectedSpanningTree and IsUndirectedSpanningForest are introduced.

### Altered Behaviour in Version 0.7.0

• The behaviour of IsSubdigraph is changed in the case that it is given one or two multidigraphs.
• The one-argument version of the operation DigraphColouring (and its synonym, DigraphColoring) is now an attribute.

## Version 0.6.1 (released 01/03/2017)

This is a minor release. This release fixes a bug in AsDigraph for a transformation and an integer. The operations OutNeighboursCopy and OutNeighborsCopy are renamed to OutNeighboursMutableCopy and OutNeighborsMutableCopy, respectively, and new operations InNeighboursMutableCopy and InNeighborsMutableCopy are introduced.

## Version 0.6.0 (released 09/12/2016)

This is a major release, adding a variety of new operations and attributes for Digraphs, as well as improving some functions and improving the package’s documentation. In this version, we welcome Luke Elliott and Markus Pfeiffer as new authors.

### New Features in Version 0.6.0

• The ability to label the edges of digraphs is introduced. [Markus Pfeiffer]
• The operation CompleteMultipartiteDigraph is introduced. [Stuart Burrell and Wilf A. Wilson]
• The operations ReadDIMACSDigraph and WriteDIMACSDigraph are introduced. [Wilf A. Wilson]
• The operation ChromaticNumber is introduced. [James D. Mitchell and Wilf A. Wilson]
• The operations IsDirectedTree and IsUndirectedTree are introduced. [Luke Elliott]
• The operation IsEulerianDigraphis introduced. [Luke Elliott]

## Version 0.5.2 (released 20/06/2016)

This is a minor release containing one bugfix and several other minor changes. Digraphs now works when it and GAP are compiled in 32 bit mode.

## Version 0.5.1 (released 08/06/2016)

This release contains some bugfixes, some minor new features, and some performance improvements. The package has moved to GitHub and we welcome Finn Smith as an author.

This release contains a new technique for encoding a vertex-coloured digraph as a vertex-coloured (undirected) graph while preserving the automorphism group, in order to calculate the automorphism group using bliss. These changes were made by Finn Smith. The previous technique involved adding two intermediate vertices for every edge; on a digraph with n vertices this could add 2n(n-1) new vertices. The new technique encodes a digraph with n vertices as a graph with 3n vertices. In certain cases, this can lead to a dramatic improvement in the time taken to calculate the automorphism group.

The new reduction is based on two techniques in:

David Bremner, Mathieu Dutour Sikiric, Dmitrii V. Pasechnik, Thomas Rehn, Achill Schürmann. Computing symmetry groups of polyhedra. https://arxiv.org/abs/1210.0206v3

Namely, “Dealing with digraphs” followed by “Reduction by superposition”. From the graph obtained by these techniques, n vertices which are irrelevant to automorphism finding are removed.

The actual reduction used is as follows: Given a digraph D=(V=[]1 .. n],E,c), create three copies V1, V2, V3 of the vertex set V. Colour V1 according to the colouring c of D, and V2, V3 with two distinct colours that do not occur in D. Connect each vertex in V1 to the corresponding vertices in V2, V3. For every arc (x,y) in E, put an edge between the copy of x in V2, and the copy of y in V3. Automorphisms of this graph, when restricted to V, are precisely the automorphisms of D. Because this changes the graph used to calculate automorphisms, the results sometimes nominally differ from the previous version - especially in the case of canonical labelling, where unrecognisably different results may appear. Generators also often appear in different orders.

The performance improvements are most noticeable on large, quite dense digraphs. On random digraphs with 5000 vertices and 0.5 edge probability, 200-400x speedups were common. When calculating the automorphism group of the complete digraph on 1000 vertices, with vertex i having colour i mod 2 + 1, we obtain a 66x speedup. When the vertex i is assigned colour i mod 7 + 1, this becomes a 400x speedup.

Minor changes include:

• a better method for DigraphReverse [Wilf A. Wilson]
• automorphism groups of complete, empty, cycle, chain, and complete bipartite digraphs are set at creation [Michael Young]
• a minor improvement in performance in the DigraphMaximalCliques [Wilf A. Wilson]
• a new operation AdjacencyMatrixMutableCopy [James D. Mitchell]

## Version 0.5.0 (released 03/03/2016)

This release contains some bugfixes, as well as new and changed functionality. Digraphs now requires the Orb package, version 4.7.5 or higher.

### New Features in Version 0.5.0

• DigraphFile and IteratorFromDigraphFile are introduced. [James D. Mitchell]
• WriteDigraphs and ReadDigraphs can now take a file as a first argument. [James D. Mitchell]
• The operation DigraphPath is introduced to find a path between two vertices in a digraph. [Wilf A. Wilson]
• The operation IteratorOfPaths is introduced to iterate over the paths between two vertices in a digraph. [Wilf A. Wilson]
• The property IsCompleteBipartiteDigraph is introduced. [Wilf A. Wilson]

### Issues Resolved in Version 0.5.0

Several bugs related to clique finding have been resolved. [Wilf A. Wilson]

• Files with extension bz2 were previously not (un)compressed when used with ReadDigraphs and WriteDigraphs. [James D. Mitchell]
• The documentation in Chapter 8 “Finding cliques and independent sets” has been corrected to accurately reflect the functionality of the package.
• A bug which led to too few cliques and independent sets being found for some digraphs has been resolved.
• A bug which led to duplicate cliques and independent sets being found for some digraphs has been resolved.

## Version 0.4.2 (released 28/01/2016)

This is a minor release to fix a bug in DigraphAllSimpleCircuits that failed to return all simple circuits in some cases Issue 13. Some documentation was also updated.

## Version 0.4.1 (released 22/01/2016)

This is a very minor release to change the version of GAP required.

## Version 0.4.0 (released 19/01/2016)

This is a major release, primarily aimed at incorporating more of the functionality of Grape into Digraphs, as well as fixing some bugs. In this version, we welcomed Jan De Beule to the development team.

### New Features in Version 0.4.0

• Functionality to calculate cliques and independent sets
• New methods for the constructor function Digraph
• ReadDigraphs and WriteDigraphs now have a new output format .p or .pickle, which allows known automorphisms to be stored with the digraph
• The following functions now use known automorphisms:
• DigraphDiameter
• DigraphDual
• DigraphShortestDistances
• Digraph method for a Grape graph
• Graph
• InDegreeSequence
• OutDegreeSequence
• The following new functions were added:
• BipartiteDoubleDigraph
• BooleanAdjacencyMatrix
• CayleyDigraph
• DigraphAddAllLoops
• DigraphAddEdgeOrbit
• DigraphAdjacencyFunction
• DigraphColoring for a digraph
• DigraphDegeneracyOrdering
• DigraphDegeneracy
• DigraphDistanceSet
• DigraphGirth
• DigraphGroup
• DigraphLayers
• DigraphLongestSimpleCircuit
• DigraphOrbitReps
• DigraphOrbits
• DigraphRemoveEdgeOrbit
• DigraphSchreierVector
• DigraphShortestDistance
• DigraphStabilizer
• DigraphUndirectedGirth
• DistanceDigraph
• DoubleDigraph
• EdgeOrbitsDigraph
• InDegreeSet
• IsDigraphEdge for a digraph and a list
• IsDistanceRegularDigraph
• IsInRegularDigraph
• IsOutRegularDigraph
• IsRegularDigraph
• JohnsonDigraph
• LineDigraph
• LineUndirectedDigraph
• MaximalSymmetricSubdigraphWithoutLoops
• MaximalSymmetricSubdigraph
• OutDegreeSet
• RepresentativeOutNeighbours

## Version 0.3.2 (released 14/01/2016)

This is another minor release due to some missing build files in the Version 0.3.1 archive.

## Version 0.3.1 (released 13/01/2016)

This is a minor release due to some missing build files in the Version 0.3 archive.

## Version 0.3.0 (released 12/01/2016)

This release contains a number of bugfixes and performance improvements.

### New Features in Version 0.3.0

• The attribute DigraphAllSimpleCircuits based on the algorithm in this paper by Donald B. Johnson. [Stuart Burrell and Wilf A. Wilson]
• Improve efficiency of the algorithm for coloring a graph with 2 colours, a method for IsBipartiteDigraph and DigraphBicomponents. [Isabella Scott and Wilf A. Wilson]
• AutomorphismGroup and DigraphCanonicalLabelling can now be used with color classes that are preserved by the permutations acting on a digraph. [James D. Mitchell]
• The TCodeDecoder was made more efficient. [James D. Mitchell]
• AsTransformation is introduced for digraphs in IsFunctionalDigraph. [James D. Mitchell]
• The tests and their code coverage were improved.

### Issues Resolved in Version 0.3.0

• There was a memory leak in bliss-0.73, which is fixed in the copy of bliss included with Digraphs, but not in the official release of bliss. [James D. Mitchell]
• Some bits of code that caused compiler warnings were improved. [James D. Mitchell]
• Some memory leaks were resolved in the Digraphs kernel module. [Michael Young]

## Version 0.2.0 (released 04/09/2015)

The first release.

## Version 0.1.0

Pre-release version that was not made publicly available.