Linear time recognition of Weak bisplit graphs

We introduced in [4]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 frame work for obtaining linear recognition algorithms for Weak bisplit graphs as well as for two sub families of this class the P-6 free bipartite graphs and the bicographs 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