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 {H1, . . . ,Hp}, it is said to be (H1, . . . ,Hp)-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 P4. We study the (un)boundedness of the clique-width of graph classes defined by two forbidden induced subgraphs H1 and H2. 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 (H1,H2)-free graphs (i) for all pairs (H1,H2), both of which are connected, except two non-equivalent cases, and (ii) for all pairs (H1,H2), 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 {H1, . . . ,Hp} 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 (H1,H2)-free graphs.