Maximum Biclique for Star 1,2,3 -free and Bounded Bimodularwidth Twin-free Bipartite Graphs $\star$

There are three usual definitions of a maximum bipartite clique (biclique) in a bipartite graph\,: either maximizing the number of vertices, or of edges, or finding a maximum balanced biclique. The first problem can be solved in polynomial time, the last ones are NP-complete. Here we show how these three problems may be efficiently solved for two classes of bipartite graphs: Star123-free twin-free graphs, and bounded bimodularwidth twin-free graphs, a class that may be defined using bimodular decomposition. Our computation requires O(n^2) time and requires a decomposition is provided, which takes respectively O(n + m) and O(mn^3) time.