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. (C) 2015 Elsevier B.V. All rights reserved.