The Complexity of Modular Graph AutomorphismV

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