GRAPE provides a basic interface to B.D. McKay's nauty (Version 2.0b5) package for calculating automorphism groups of (possibly vertex-coloured) graphs and for testing graph isomorphism (see Nau90). To use functions depending on nauty, GRAPE must be fully installed on a computer running UNIX (see Installing the GRAPE Package).
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
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 canonical labelling is used, which was the behaviour in versions
of GRAPE before 4.0. However, since canonical labellings can depend on
the version of nauty (currently 2.0b5), certain parameters of nauty
(always set the same for a given graph by GRAPE 4.1), 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. We remark that GRAPE 4.1 now
sets nauty parameters differently than before for non-simple graphs
(and so canonical labellings may be different than before), in order to
avoid a performance problem with certain sparse directed graphs.
gap> gamma := JohnsonGraph(7,4);; gap> delta := JohnsonGraph(7,3);; gap> IsIsomorphicGraph( gamma, delta ); true
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