Graph Isomorphism for Graph Classes Characterized by Two Forbidden Induced Subgraphs

Abstract

We study the complexity of the Graph Isomorphism problem on graph classes that are characterized by a finite number of forbidden induced subgraphs, focusing mostly on the case of two forbidden subgraphs. We show hardness results and develop techniques for the structural analysis of such graph classes, which applied to the case of two forbidden subgraphs give the following results: A dichotomy into isomorphism complete and polynomial-time solvable graph classes for all but finitely many cases, whenever neither of the forbidden graphs is a clique, a pan, or a complement of these graphs. Further reducing the remaining open cases we show that (with respect to graph isomorphism) forbidding a pan is equivalent to forbidding a clique of size three.

Induced Minor Free Graphs: Isomorphism and Clique-Width

“…The classes of co-bipartite graphs and restricted split graphs are closed under vertex deletions and edge contractions, and thus under induced minors. As also argued in [31] and [23], the standard graph-isomorphism reductions to split graphs and co-bipartite graphs 1 explained in [4] imply the following lemma.…”

“…When forbidding one induced subgraph, it is known that Graph Isomorphism can be solved in polynomial time on H -free graphs if H is an induced subgraph of P 4 (the path on four vertices) and is GI-complete otherwise (see [4]). For two forbidden induced subgraphs such a classification into GI-complete and polynomial-time solvable cases turns out to be more complicated [23,33]. In the case where we consider forbidden subgraphs (i.e., also allowing edge deletions) there is a complete dichotomy for the computational complexity of Graph Isomorphism on classes characterized by a finite set of forbidden subgraphs, while there are intermediate classes defined by infinitely many forbidden subgraphs for which the problem is neither polynomial-time solvable nor GI-complete [27] (assuming that graph isomorphism is not polynomial-time solvable).…”

Bounding the clique-width of H-free split graphs

“…Similar studies have been performed for variants of clique-width, such as linear clique-width [26] and power-bounded clique-width [2]. Moreover, the (un)boundedness of the cliquewidth of a graph class seems to be related to the computational complexity of the Graph Isomorphism problem, which has in particular been investigated for graph classes defined by two forbidden induced subgraphs [28,33]. Indeed, a common technique (see e.g.…”

List coloring in the absence of two subgraphs

“…We consider a more general version of graph coloring called list coloring and classify the complexity of this problem for graphs characterized by two forbidden induced subgraphs. Kratsch and Schweitzer [26] and Lozin [27] performed a similar study as ours for the problems graph isomorphism and dominating set, respectively. Before we summarize related coloring results and explain our new results, we first state the necessary terminology.…”