Classifying the Clique-Width of H-Free Bipartite Graphs

Let G be a bipartite graph, and let H be a bipartite graph with a fixed bipartition (BH ,WH). We consider three different, natural ways of forbidding H as an induced subgraph in G. First, G is H-free if it does not contain H as an induced subgraph. Second, G is strongly H-free if G is H-free or else has no bipartition (BG,WG) with BH ⊆ BG and WH ⊆ WG. Third, G is weakly H-free if G is H-free or else has at least one bipartition (BG,WG) with BH 6⊆ BG or WH 6⊆WG. Lozin and Volz characterized all bipartite graphs H for which the class of strongly Hfree bipartite graphs has bounded clique-width. We extend their result by giving complete classifications for the other two variants of H-freeness.