Yet Another Reduction from Graph to Ring Isomorphism Problems

It has been known that the graph isomorphism problem is polynomial-time many-one reducible to the ring isomorphism problem. In fact, two different reductions have already been proposed. For those reductions, rings of certain types have been used to represent a given graph. In this paper, we give yet another reduction, which is based on a simpler and more natural construction of a ring from a graph. By the existing reductions, one of the original graph isomorphisms can be found in each ring isomorphism obtained for the reduced ring isomorphism problem instance. On the other hand, in our new reduction, it is not clear how to get a graph isomorphism between two graphs from an obtained ring isomorphism between rings constructed from the graphs. However, we show that we can compute a graph isomorphism from an obtained ring isomorphism in polynomial time. In fact, one ring isomorphism may correspond to many graph isomorphisms in our reduction. Our proof essentially shows a way to obtain all graph isomorphisms corresponding to one ring isomorphism.