The Complexity of Modular Graph Automorphism
V. Arvind, Richard Beigel, Antoni Lozano · SIAM Journal on Computing · 2000
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 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 difficulty between graph automorphism and graph isomorphism. We define an appropriate search problem corresponding to 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 $\forall k>1$.