The Complexity of Modular Graph Automorphism

Motivated by the question of the relative complexities of the Graph Isomorphism and the Graph Automorphism problems we de ne and study the modular graph automor phism problems These are the decision problems modk GA which consist for each k of deciding whether the number of automorphisms of a graph is divisible by k The modk GA problems all turn out to be intermediate in di culty between Graph Automorphism and Graph Isomorphism We de ne an appropriate search version of modk GA and design an algorithm that polynomial time reduces the modk GA search problem to the decision problem Com bining this algorithm with an IP protocol we obtain a randomized polynomial time checker for modk GA for all k