Classifying the clique-width of HH-free bipartite graphs

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