An O(n3 log n) deterministic and an O (n 3) probabilistic isomorphism test for trivalent graphs

The main results of this paper are an O(n3) probabilistic algorithm and an O(n3 log n) deterministic algorithm that test whether two given trivalent graphs are isomorphic. In fact, the algorithms construct the set of all isomorphisms of the two graphs. Variants of these algorithms construct the set of all automorphisms of a trivalent graph. The algorithms make use of some new improved permutation group algorithms that exploit the fact that the groups involved are 2-groups. A remarkable property of the probabilistic algorithm is that it computes Isoe,ei(X,Y), i = 1,...,m, m = O(n) (the set of all isomorhisms φ: X → Y with φ(e)=ei) for the cost of computing the single set Isoe,el(X,Y).