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

• Authors: Alain Bretto;Alain Faisant;Thierry Vallée
• Venue: Theoretical Computer Science
• Year: 2006

In one hand the graph isomorphism problem (GI) has received considerable attention due to its unresolved complexity status and is many practical applications. On the other hand a notion of compatible topologies on graphs has emerged from digital topology (see [A. Bretto, Comparability graphs and digital topology, Comput. Vision Graphic Image Process. (Image Understanding), 82 (2001) 33-41; J.M. Chassery, Connectivity and consecutivity in digital pictures, Comput. Vision Graphic Image Process. 9 (1979) 294-300; L.J. Latecki, Topological Connectedness and 8-connectness in digital pictures, CVGIP Image Understanding 57(2) (1993) 261-262; U. Eckhardt, L.J. Latecki, Topologies for digital spaces Z2 and Z3, Comput. Vision Image Understanding 95 (2003) 261-262; T.Y. Kong, R. Kopperman, P.R. Meyer, A topological approach to digital topology, Amer. Math. Monthly Archive 98(12) (1991) 901-917; R. Kopperman, Topological digital topology, Discrete geometry for computer imagery, 11th International Conference, Lecture Notes in Computer Science, Vol. 2886, DGCI 2003, Naples, Italy, November 19-21, pp. 1-15]).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.