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 ( B H , W H ) . 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 no bipartition of? G contains an induced copy of? H in a way that respects the bipartition of? H . Third, G is weakly H -free if? G has at least one bipartition that does not contain an induced copy of? H in a way that respects the bipartition of? H . Lozin and Volz characterized all bipartite graphs? H for which the class of strongly H -free bipartite graphs has bounded clique-width. We extend their result by giving complete classifications for the other two variants of H -freeness.

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