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

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.

[1]  Seymour Lipschutz,et al.  Schaum's outline of theory and problems of general topology , 1965 .

[2]  Rudolf Mathon,et al.  A Note on the Graph Isomorphism counting Problem , 1979, Inf. Process. Lett..

[3]  Ralph Kopperman,et al.  A Jordan surface theorem for three-dimensional digital spaces , 1991, Discret. Comput. Geom..

[4]  H. Herrlich,et al.  Category theory at work , 1991 .

[5]  Leila Schneps The Grothendieck theory of dessins d'enfants: Dessins d'enfants on the Riemann sphere , 1994 .

[6]  Ralph Kopperman,et al.  Topological Digital Topology , 2003, DGCI.

[7]  A. Blumberg BASIC TOPOLOGY , 2002 .

[8]  Gary L. Miller Graph Isomorphism, General Remarks , 1979, J. Comput. Syst. Sci..

[9]  Longin Jan Latecki,et al.  Topological connectedness and 8-connectedness in digital pictures , 1993 .

[10]  R. E. Stong,et al.  Finite topological spaces , 1966 .

[11]  Alain Bretto Comparability Graphs and Digital Topology , 2001, Comput. Vis. Image Underst..

[12]  Ulrich Eckhardt,et al.  Topologies for the digital spaces Z2 and Z3 , 2003, Comput. Vis. Image Underst..

[13]  Kellogg S. Booth,et al.  Isomorphism Testing for Graphs, Semigroups, and Finite Automata Are Polynomially Equivalent Problems , 1978, SIAM J. Comput..

[14]  John E. Hopcroft,et al.  Linear time algorithm for isomorphism of planar graphs (Preliminary Report) , 1974, STOC '74.

[15]  Jean-Marc Chassery,et al.  Connectivity and consecutivity in digital pictures , 1979 .

[16]  J. Köbler,et al.  The Graph Isomorphism Problem: Its Structural Complexity , 1993 .

[17]  Reinhard Diestel,et al.  Directions in infinite graph theory and combinatorics , 1992 .

[18]  Allen R. Hanson,et al.  Dynamic mutual calibration and view planning for cooperative mobile robots with panoramic virtual stereo vision , 2004, Comput. Vis. Image Underst..

[19]  T. Yung Kong,et al.  A topological approach to digital topology , 1991 .