The graph isomorphism problem on geometric graphs

Ryuhei Uehara

doi:10.46298/dmtcs.2076

Abstract

Special issue PRIMA 2013 The graph isomorphism (GI) problem asks whether two given graphs are isomorphic or not. The GI problem is quite basic and simple, however, it\textquoterights time complexity is a long standing open problem. The GI problem is clearly in NP, no polynomial time algorithm is known, and the GI problem is not NP-complete unless the polynomial hierarchy collapses. In this paper, we survey the computational complexity of the problem on some graph classes that have geometric characterizations. Sometimes the GI problem becomes polynomial time solvable when we add some restrictions on some graph classes. The properties of these graph classes on the boundary indicate us the essence of difficulty of the GI problem. We also show that the GI problem is as hard as the problem on general graphs even for grid unit intersection graphs on a torus, that partially solves an open problem.

Highlights

For any two given graphs G1 = (V1, E1) and G2 = (V2, E2) with |V1| = |V2|, the graph isomorphism (GI) problem asks whether there exists a one-to-one mapping φ between two given graphs
We show a hierarchy of graph classes with respect to the computational complexity of the GI problem
We give a partial answer to the unit grid intersection graphs: it is GI-complete if they are on a torus

Summary

Introduction

From the viewpoint of the computational complexity, it is not likely that the GI problem is NP-complete. (We note that this property does not hold on the recognition problem for these graph classes. This fact will be discussed in the concluding remarks.) The hereditary property leads us to the notion of GI-completeness: The GI problem on the class C is said to be GI-complete if it is as hard as on general graphs under polynomial time reduction (see, e.g., Uehara et al (2004)). In 1970s, some efficient algorithms were developed for the GI problem on basic graph classes, which include planar graphs (Hopcroft and Tarjan (1974)), interval graphs (Booth and Lueker (1976)). To demonstrate some basic techniques, we prove that the GI problem is GI complete for unit grid intersection graphs on a torus, which partially solves an open problem

Preiminalies

Geometric graph classes and their relationship

GI completeness of unit grid intersection graph on a torus

Concluding Remarks