Generic case complexity of the Graph Isomorphism Problem

Abstract The edge test is a partial algorithm for the Graph Isomorphism Problem based on comparison the number of edges. We perform a probabilistic analysis of the efficiency of the edge test. With the binomial distribution B(n,p) on the set of inputs, we estimate the asymptotic failure probability of the edge test depending on the rate of decay of parameter p. In particular, if p ≤ 1/2, np → λ > 0, then the asymptotic failure probability is nonzero, so that the edge test does not solve generically the Graph Isomorphism Problem. On the other hand, if p ≤ 1/2, np → ∞, then the failure set is negligible and the edge test generically solves the Graph Isomorphism Problem in polynomial time.