[grape] 8 Automorphism groups and isomorphism testing for graphs

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# 8 Automorphism groups and isomorphism testing for graphs

### Sections

1. AutGroupGraph
2. IsIsomorphicGraph
3. GraphIsomorphismClassRepresentatives
4. GraphIsomorphism

GRAPE provides a basic interface to B.D. McKay's nauty (Version 2.2 final) package for calculating automorphism groups of (possibly vertex-coloured) graphs and for testing graph isomorphism (see Nau90). To use a function described in this chapter, which depends on nauty, GRAPE must be fully installed on a computer running UNIX (see Installing the GRAPE Package).

## 8.1 AutGroupGraph

- `AutGroupGraph(` gamma`)`
- `AutGroupGraph(` gamma`,` colourclasses`)`

  The first version of this function returns the automorphism group of the (directed) graph gamma, using nauty (this can also be accomplished by typing `AutomorphismGroup(`gamma`)`). The automorphism group Aut(gamma) of gamma is the group consisting of the permutations of the vertices of gamma which preserve the edge-set of gamma.

  In the second version, colourclasses is an ordered partition of the vertices of gamma (into colour-classes), and the subgroup of Aut(gamma) preserving this ordered partition is returned. The ordered partition should be given as a list of sets, although the last set in the list may be omitted. Note that we do not require that adjacent vertices be in different colour-classes.

      gap> gamma := JohnsonGraph(4,2);
      rec( isGraph := true, order := 6,
        group := Group([ (1,4,6,3)(2,5), (2,4)(3,5) ]),
        schreierVector := [ -1, 2, 1, 1, 1, 1 ], adjacencies := [ [ 2, 3, 4, 5 ] ],
        representatives := [ 1 ],
        names := [ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 2, 3 ], [ 2, 4 ], [ 3, 4 ] ],
        isSimple := true )
      gap> Size(AutGroupGraph(gamma));
      48
      gap> Size(AutGroupGraph(gamma,[[1,2,3],[4,5,6]]));
      6
      gap> Size(AutGroupGraph(gamma,[[1,6]]));
      16

  ## 8.2 IsIsomorphicGraph

- `IsIsomorphicGraph(` gamma1`,` gamma2`)`
- `IsIsomorphicGraph(` gamma1`,` gamma2`,` firstunbindcanon`)`

  This boolean function uses the nauty package to test whether graphs gamma1 and gamma2 are isomorphic. The value `true` is returned if and only if the graphs are isomorphic (as directed, uncoloured graphs).

  The optional boolean parameter firstunbindcanon determines whether or not the `canonicalLabelling` components of both gamma1 and gamma2 are first unbound before testing isomorphism. If firstunbindcanon is `true` (the default, safe and possibly slower option) then these components are first unbound. If firstunbindcanon is `false`, then any existing `canonicalLabelling` components are used, which was the behaviour in versions of GRAPE before 4.0. However, since canonical labellings can depend on the version of nauty, the version of GRAPE, parameter settings of nauty, and the compiler and computer used, you must be sure that if firstunbindcanon=`false` then the `canonicalLabelling` component(s) which may already exist for gamma1 or gamma2 were created in exactly the same environment in which you are presently computing.

  See also GraphIsomorphism. For pairwise isomorphism testing of three or more graphs, see GraphIsomorphismClassRepresentatives.

      gap> gamma := JohnsonGraph(7,4);;
      gap> delta := JohnsonGraph(7,3);;
      gap> IsIsomorphicGraph( gamma, delta );
      true

  ## 8.3 GraphIsomorphismClassRepresentatives

- `GraphIsomorphismClassRepresentatives(` L`)`
- `GraphIsomorphismClassRepresentatives(` L`,` firstunbindcanon`)`

  Given a list L of graphs, this function uses nauty to return a list consisting of pairwise non-isomorphic elements of L, representing all the isomorphism classes of elements of L.

  The optional boolean parameter firstunbindcanon determines whether or not the `canonicalLabelling` components of all elements of L are first unbound before proceeding. If firstunbindcanon is `true` (the default, safe and possibly slower option) then these components are first unbound. If firstunbindcanon is `false`, then any existing `canonicalLabelling` components of elements of L are used. However, since canonical labellings can depend on the version of nauty, the version of GRAPE, parameter settings of nauty, and the compiler and computer used, you must be sure that if firstunbindcanon=`false` then the `canonicalLabelling` component(s) which may already exist for elements of L were created in exactly the same environment in which you are presently computing.

      gap> A:=JohnsonGraph(5,2);
      rec( isGraph := true, order := 10,
        group := Group([ (1,5,8,10,4)(2,6,9,3,7), (2,5)(3,6)(4,7) ]),
        schreierVector := [ -1, 2, 2, 1, 1, 1, 2, 1, 1, 1 ],
        adjacencies := [ [ 2, 3, 4, 5, 6, 7 ] ], representatives := [ 1 ],
        names := [ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 5 ], [ 2, 3 ], [ 2, 4 ],
            [ 2, 5 ], [ 3, 4 ], [ 3, 5 ], [ 4, 5 ] ], isSimple := true )
      gap> B:=JohnsonGraph(5,3);
      rec( isGraph := true, order := 10,
        group := Group([ (1,7,10,6,3)(2,8,4,9,5), (4,7)(5,8)(6,9) ]),
        schreierVector := [ -1, 1, 1, 2, 1, 1, 1, 2, 1, 1 ],
        adjacencies := [ [ 2, 3, 4, 5, 7, 8 ] ], representatives := [ 1 ],
        names := [ [ 1, 2, 3 ], [ 1, 2, 4 ], [ 1, 2, 5 ], [ 1, 3, 4 ], [ 1, 3, 5 ],
            [ 1, 4, 5 ], [ 2, 3, 4 ], [ 2, 3, 5 ], [ 2, 4, 5 ], [ 3, 4, 5 ] ],
        isSimple := true )
      gap> R:=GraphIsomorphismClassRepresentatives([A,B,ComplementGraph(A)]);;
      gap> Length(R);
      2
      gap> List(R,VertexDegrees);
      [ [ 6 ], [ 3 ] ]

  ## 8.4 GraphIsomorphism

- `GraphIsomorphism(` gamma1`,` gamma2`)`
- `GraphIsomorphism(` gamma1`,` gamma2`,` firstunbindcanon`)`

  If graphs gamma1 and gamma2 are isomorphic, then this function uses nauty to return an isomorphism from gamma1 to gamma2. This isomorphism will be a permutation of `[1..`gamma1`.order]` which maps the edge-set of gamma1 to that of gamma2. If gamma1 and gamma2 are not isomorphic then this function returns `fail`.

  The optional boolean parameter firstunbindcanon determines whether or not the `canonicalLabelling` components of both gamma1 and gamma2 are first unbound before proceeding. If firstunbindcanon is `true` (the default, safe and possibly slower option) then these components are first unbound. If firstunbindcanon is `false`, then any existing `canonicalLabelling` components are used. However, since canonical labellings can depend on the version of nauty, the version of GRAPE, parameter settings of nauty, and the compiler and computer used, you must be sure that if firstunbindcanon=`false` then the `canonicalLabelling` component(s) which may already exist for gamma1 or gamma2 were created in exactly the same environment in which you are presently computing.

  See also IsIsomorphicGraph.

      gap> A:=JohnsonGraph(5,2);
      rec( isGraph := true, order := 10,
        group := Group([ (1,5,8,10,4)(2,6,9,3,7), (2,5)(3,6)(4,7) ]),
        schreierVector := [ -1, 2, 2, 1, 1, 1, 2, 1, 1, 1 ],
        adjacencies := [ [ 2, 3, 4, 5, 6, 7 ] ], representatives := [ 1 ],
        names := [ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 5 ], [ 2, 3 ], [ 2, 4 ],
            [ 2, 5 ], [ 3, 4 ], [ 3, 5 ], [ 4, 5 ] ], isSimple := true )
      gap> B:=JohnsonGraph(5,3);
      rec( isGraph := true, order := 10,
        group := Group([ (1,7,10,6,3)(2,8,4,9,5), (4,7)(5,8)(6,9) ]),
        schreierVector := [ -1, 1, 1, 2, 1, 1, 1, 2, 1, 1 ],
        adjacencies := [ [ 2, 3, 4, 5, 7, 8 ] ], representatives := [ 1 ],
        names := [ [ 1, 2, 3 ], [ 1, 2, 4 ], [ 1, 2, 5 ], [ 1, 3, 4 ], [ 1, 3, 5 ],
            [ 1, 4, 5 ], [ 2, 3, 4 ], [ 2, 3, 5 ], [ 2, 4, 5 ], [ 3, 4, 5 ] ],
        isSimple := true )
      gap> GraphIsomorphism(A,B);
      (3,4,7,8,6,5)
      gap> GraphIsomorphism(A,ComplementGraph(A));
      fail

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