On the nonuniform complexity of the graph isomorphism problem

The nonuniform complexity of the graph isomorphism (GI) and graph automorphism (GA) problems is studied, and the implications of different types of polynomial-time reducibilities from these problems to sparse sets are considered. It is shown that if GI (or GA) is bounded truth-table or conjunctively reducible to a sparse set, then it is in P; if it is supposed that it is truth-table reducible without restrictions to a sparse set (or, equivalently, that it belongs to P/poly), then the problem is low for MA, the class of sets with publishable proofs. With respect to nondeterministic reductions, it is shown that if the considered problems are btt strong nondeterministically reducible to a sparse set, then they are in co-NP. Some of these results are proved using graph constructions that show properties of the GI and GA problems.< >