Isomorphism of graphs of bounded valence can be tested in polynomial time

Abstract

Suppose we are given a set of generators for a group G of permutations of a colored set A. The color automorphism problem for G involves finding generators for the subgroup of G which stabilizes the color classes. It is shown that testing isomorphism of graphs of bounded valance is polynomial-time reducible to the color automorphism problem for groups with composition factors of bounded order. The algorithm for the latter problem involves three divide-and-conquer maneuvers. The problem is solved sequentially on the G-orbits. An orbit is broken into a minimal system of imprimitivity blocks. At that point, the hypothesis on G guarantees the presence of a subgroup P of “small” index which acts a p-group on the blocks. Divide-and-conquer is then used on the group, trading the problem on G for a small number of similar problems on P. In the trivalent case p = 2, P = G and the analysis requires only elementary notions. For higher valence, the justification requires some new observations about primitive permutation groups.