Here - Combinatorial algorithms and algorithmic graph theory

20 Generation of cubic graphs
We use nauty [88] to compute the automorphism group of a graph G. Nauty
does not compute orbits of edges or pairs of edges of G under the action of
Aut(G) by default, but by using the userautomproc option in nauty, we can save
the generators of Aut(G). Using these generators we can compute the orbits of
edges or pairs of edges (e.g. by using a union-find algorithm).
The class of prime graphs is not closed under the three construction operations.
So after applying a construction operation we still have to test whether
the constructed graph is still prime. If it is not prime, the graph is not accepted
by the algorithm.
The coarse pseudocode for generating all non-isomorphic prime graphs is given
in Algorithm 2.1 (cf. Algorithm 1.1 from Section 1.3).
Algorithm 2.1 Construct prime graphs(prime graph G)
output G
if G has less vertices than the desired number of vertices then
find expansions
compute classes of equivalent expansions
for each equivalence class do
choose one expansion X
perform expansion X
if expansion is canonical and expanded graph is prime then
Construct prime graphs(expanded graph)
end if
perform reduction X −1
end for
end if