Compatible topologies on graphs: An application to graph isomorphism problem complexity

On one hand the graph isomorphism problem (GI) has received considerable attention due to its unresolved complexity status and its many practical applications. On the other hand a notion of compatible topologies on graphs has emerged from digital topology (see [5, 7, 17, 9, 14, 16]). In this article we study GI from the topological point of view. Firstly we explore the poset of compatible topologies on graphs and in particular on bipartite graphs. Then, from a graph we construct a particular compatible Alexandroff topological space said homeomorphic-equivalent to the graph. Conversely from any Alexandroff topology we construct an isomorphic-equivalent graph on which the topology is compatible. Finally using these constructions, we show that GI is polynomial-time equivalent to the topological homeomorphism problem (TopHomeo). Hence GI and TopHomeo are in the same class of complexity.