The complexity of modular graph automorphism

Abstract

Motivated by the question of the relative complexities of the Graph Isomorphism and the Graph Automorphism problems, we define and study the modular graph automorphism problems. These are the decision problems ModkGA which consist, for each k > 1, of deciding whether the number of automorphisms of a graph is divisible by k. The ModkGA problems all turn out to be intermediate in difficulty between Graph Automorphism and Graph Isomorphism. We define an appropriate search version of ModkGA and design an algorithm that polynomial-time reduces the ModkGA search problem to the decision problem. Combining this algorithm with an IP protocol, we obtain a randomized polynomial-time checker for ModkGA, for all k > 1. © 1998 Springer-Verlag.