Here - Combinatorial algorithms and algorithmic graph theory

34 Generation of cubic graphs
Figure 2.12: Reusing information about the orbits of the automorphism group of the
parent graph to speed up the computation of the automorphism group
of the expanded graph.
phism of G mapping φ(v) to φ(w).
If all vertices which are part of an extensible set S of a graph G are in a
different orbit of Aut(G), we can sometimes apply another optimisation. Let
|Γ(v)| be the size of the orbit of a v ∈ V (G) of Aut(G). If for an automorphism
γ ∈ Aut(G) and a v ∈ S it holds that γ(v) ≠ v, then this automorphism cannot
lead to an automorphism of T (G, S). If ∃ v ∈ S : |Γ(v)| = | Aut(G)|, this means
that ∀ γ ∈ Aut(G) : γ(v) ≠ v. So then the automorphism group of T (G, S)
will be trivial, thus we also do not have to call nauty to compute it. Though in
practice this optimisation only yields a tiny speedup.
2.5 Generation of non-prime graphs without reducible
triangles
2.5.1 Isomorphism rejection
Recall that we gave the triangle operation priority over the edge operation. So
in principle the non-prime connected cubic graphs without reducible triangles on
n vertices are generated by applying the edge insertion operation to each pair of
edges in a graph on n − 2 vertices that guarantees that no reducible triangles are
present in the resulting graph. We call such a pair an extensible pair of edges. In
Section 2.5.2 we go into more detail about determining which pairs of edges are
extensible.