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 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 colouring problem restricted to $$(H_1,H_2)$$ -free graphs.