The complexity of graph isomorphism for colored graphs with color classes of size 2 and 3

Abstract

We prove that the graph isomorphism problem restricted to colored graphs with color multiplicities 2 and 3 is complete for symmetric logarithmic space SL under many-one reductions.This result improves the existing upper bounds for the problem. We also show that the graph automorphism problem for colored graphs with color classes of size 2 is equivalent to deciding whether an undirected graph has more than a single connected component and we prove that for color classes of size 3 the graph automorphism problem is contained in SL.