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.
Highlights
Given two graphs G1 and G2, the graph isomorphism problem asks whether there exists a bijection from the vertices of G1 to the vertices of G2 that preserves adjacency and nonadjacency
We extend the techniques of bounded color valence and hypergraph isomorphism on hypergraphs of bounded color class size as follows
We show that isomorphism of graphs of bounded generalized color valence can be solved in polynomial time
Summary
Given two graphs G1 and G2, the graph isomorphism problem asks whether there exists a bijection from the vertices of G1 to the vertices of G2 that preserves adjacency and nonadjacency. With respect to classes defined by forbidden induced subgraphs, there is a dichotomy for the isomorphism problem on H1-free graphs into polynomially solvable and isomorphism complete cases. The applications of the techniques include for example showing that bipartite graphs that are free from a fixed forbidden double star refine into graphs of bounded generalized color valence This can be used to show that isomorphism of graphs of bounded clique number with a fixed forbidden double star can be solved in polynomial time. We devise a methodology to prove isomorphism completeness results (Section 3) and briefly describe how to simulate a complete invariant given only an isomorphism algorithm (Section 4) After this we turn to techniques for isomorphism testing using modular decompositions (Section 5) and devise a polynomial time algorithm for graphs of bounded generalized color valence (Section 6). The techniques can be applied to analyze specific triangle-free graphs and specific graphs of bounded clique number. (For details see [29].) We conclude by showing that together with the theorems in [21] this resolves the complexity of all but finitely many graph classes defined by two forbidden induced subgraphs (Section 7)
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