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Towards an Isomorphism Dichotomy for Hereditary Graph Classes

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Abstract

In this paper we resolve the complexity of the isomorphism problem on all but finitely many of the graph classes characterized by two forbidden induced subgraphs. To this end we develop new techniques applicable for the structural and algorithmic analysis of graphs. First, we develop a methodology to show isomorphism completeness of the isomorphism problem on graph classes by providing a general framework unifying various reduction techniques. Second, we generalize the concept of the modular decomposition to colored graphs, allowing for non-standard decompositions. We show that, given a suitable decomposition functor, the graph isomorphism problem reduces to checking isomorphism of colored prime graphs. Third, we extend the techniques of bounded color valence and hypergraph isomorphism on hypergraphs of bounded color class size as follows. We say a colored graph has generalized color valence at most k if, after removing all vertices in color classes of size at most k, for each color class C every vertex has at most k neighbors in C or at most k non-neighbors in C. We show that isomorphism of graphs of bounded generalized color valence can be solved in polynomial time.

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Acknowledgments

I thank Matasha Mazis for inspiring comments and suggestions.

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Authors and Affiliations

  1. RWTH Aachen University, Aachen, Germany

    Pascal Schweitzer

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  1. Pascal Schweitzer
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Correspondence to Pascal Schweitzer.

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An extended abstract of this paper was presented at STACS 2015 [35]

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Schweitzer, P. Towards an Isomorphism Dichotomy for Hereditary Graph Classes. Theory Comput Syst 61, 1084–1127 (2017). https://doi.org/10.1007/s00224-017-9775-8

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