Clique-width of Graph Classes Defined by Two Forbidden Induced Subgraphs

Abstract

If a graph has no induced subgraph isomorphic to any graph in a finite family \(\{H_1,\ldots,H_p\}\), it is said to be \((H_1,\ldots,H_p)\)-free. The class of \(H\)-free graphs has bounded clique-width if and only if \(H\) is an induced subgraph of the 4-vertex path \(P_4\). We study the (un)boundedness of the clique-width of graph classes defined by two forbidden induced subgraphs \(H_1\) and \(H_2\). Prior to our study it was not known whether the number of open cases was finite. We provide a positive answer to this question. To reduce the number of open cases we determine new graph classes of bounded clique-width and new graph classes of unbounded clique-width. For obtaining the latter results we first present a new, generic construction for graph classes of unbounded clique-width. Our results settle the boundedness or unboundedness of the clique-width of the class of \((H_1,H_2)\)-free graphs (i) for all pairs \((H_1,H_2)\), both of which are connected, except two non-equivalent cases, and (ii) for all pairs \((H_1,H_2)\), at least one of which is not connected, except 11 non-equivalent cases. We also consider classes characterized by forbidding a finite family of graphs \(\{H_1,\ldots,H_p\}\) as subgraphs, minors and topological minors, respectively, and completely determine which of these classes have bounded clique-width. Finally, we show algorithmic consequences of our results for the graph colour