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

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 unboundedness 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 graphsifor all pairs $$H_1,H_2$$, both of which are connected, except two non-equivalent cases, andiifor all pairs $$H_1,H_2$$, at least one of which is not connected, excepti?ź11 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 colouring problem restricted to $$H_1,H_2$$-free graphs.