This paper describes an O ( n 3 log n ) deterministic algorithm and an O ( n 3 ) Las Vegas algorithm for testing whether two given trivalent graphs on n vertices are isomorphic. In fact, the algorithms construct the set of all isomorphisms between two such graphs, presenting, in particular, generators for the group of all automorphisms of a trivalent graph. The algorithms are based upon the original polynomial-time solution to these problems by Luks but they introduce numerous speedups. These include improved permutation-group algorithms that exploit the structure of the underlying 2-groups. A remarkable property of the Las Vegas algorithm is that it computes the set of all isomorphisms between two trivalent graphs for the cost of computing only those isomorphisms that map a specified edge to a specified edge.

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).