We describe here a technique of decomposition of bipartite graphs which seems to be as interesting within this context as the well known modular and split techniques for the decomposition of general graphs. In particular, we characterize by forbidden subgraphs the family of bipartite graphs which are totally decomposable (i.e. reducible to single vertices) with respect to our decomposition. This family contains previously known families of graphs such as bicographs and P6-free bipartite graphs. As an application we provide polynomial solutions of optimization problems, some of them being NP-complete for general bipartite graphs.

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We introduce a new family of bipartite graphs which is the bipartite analogue of the class of complement reducible graphs or cographs. A bi-complement reducible Ž . graph or bi-cograph is a bipartite graph G s W j B, E that can be reduced to single vertices by recursively bi-complementing the edge set of all connected bip bipartite subgraphs. The bi-complemented graph G of G is the graph having the same vertex set W j B as G, while its edge set is equal to W = B y E. The aim of this paper is to show that there exists an equivalent definition of bi-cographs by three forbidden configurations. We also propose a tree representation for this class of graphs. ᮊ 1997 Academic Press

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