Abstract
Abstract We introduced in [5] a new decomposition scheme for bipartite graphs that we called canonical decomposition. Weak-bisplit graphs are totally decomposable following this decomposition. We give here the framework for obtaining linear recognition algorithms for Weak-bisplit graphs as well as for two subfamilies of this class, the P 6 -free bipartite graphs and the bi-cographs. We conclude by presenting a list of NP-complete optimization problems for general bipartite graphs which they have efficient solutions when dealing with weak-bisplit graphs.
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