Here - Combinatorial algorithms and algorithmic graph theory

36 Generation of cubic graphs
The discriminating power of x 0 , . . . , x 4 is sufficient to avoid the more expensive
computation of x 5 , x 6 in 92% of the cases for n = 26. This fraction is increasing
for larger values of n.
The values x 0 , . . . , x 4 are invariant under isomorphisms, so edges that are
equivalent under the automorphism group have the same values. The values x 5 , x 6
have an even stronger property: two edges are equivalent under the automorphism
group if and only if x 5 , x 6 are the same. So two edges have the same tuple
(x 0 , . . . , x 6 ) if and only if they are in the same orbit of the automorphism group
of the graph. Together with the definition of canonical labelling, this implies the
following:
Lemma 2.6. Let G 1 and G 2 be connected cubic graphs with reducible edges, and
let γ be an isomorphism from G 1 to G 2 . Let e 1 and e 2 be, respectively, reducible
edges of G 1 and G 2 having largest 7-tuples. Then γ(e 1 ) is in the same orbit as
e 2 . Furthermore, the graph Ḡ1 obtained by reducing e 1 in G 1 is isomorphic to
the graph Ḡ2 obtained by reducing e 2 in G 2 .
Next to that, there is an isomorphism from Ḡ1 to Ḡ2 mapping the edge pairs
to which e 1 and e 2 were reduced onto each other: Let p 1 be the edge pair of Ḡ1
obtained by applying the edge reduction to e 1 in G 1 and let p 2 be the edge pair
of Ḡ2 obtained by applying the edge reduction to e 2 in G 2 . Restricting the isomorphism
from G 1 to G 2 that maps e 1 to e 2 to the vertices that already belonged
to Ḡ1 and Ḡ2, respectively, yields an isomorphism from Ḡ1 to Ḡ2 mapping p 1 to
p 2 .
The algorithm would work correctly if only x 5 , x 6 were computed, but x 0 , . . . , x 4
are important for the efficiency of the algorithm. While for x 5 , x 6 a canonical
form must be computed, the earlier values are based on purely local criteria that
are cheaper to compute. Furthermore these criteria allow some look-ahead. For
example, when expanding a graph G by inserting a new edge, it is easy to decide
already on the level of G whether x 0 will be 1 or 0 for this edge in the expanded
graph G ′ and whether other edges with x 0 = 1 in G ′ will exist. This allows us
to avoid the construction of a lot of children that would afterwards be rejected
because the last edge inserted does not have maximal value of (x 0 , . . . , x 6 ). We
go into more detail about this in Section 2.5.2.
What we still have to make sure is that from a graph with n − 2 vertices
we never construct two isomorphic graphs that are both accepted. Therefore,
when applying the edge insertion operation to a graph G, we first determine its
automorphism group Aut(G). After constructing the set of all extensible edge