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.

The notion of a δ-minimum edge-colouring was introduced by J-L. Fouquet (in his french PhD Thesis [2]). Here we present some structural properties of δ-minimum edge-colourings, partially taken from the above thesis. The paper serves as an auxiliary tool for another paper submitted by the authors to Graphs and Combinatorics.

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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In [7] was introduced a new decomposition scheme for bipartite graphs that was called canonical decomposition. Weak-bisplit graphs are totally decomposable following this decomposition. We give here linear time algorithms for the recognition of weak-bisplit graphs as well as for two subclasses of this class, the P6-free bipartite graphs and the bi-cographs. Our algorithms extends the technics developped in [2] for cographs's recognition. We conclude by presenting efficient solutions for some optimization problems when dealing with weak-bisplit graphs.

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If G is a bridgeless cubic graph, Fulkerson conjectured that we can find 6 perfect matchings (a Fulkerson covering) with the property that every edge of G is contained in exactly two of them. A consequence of the Fulkerson conjecture would be that every bridgeless cubic graph has 3 perfect matchings with empty intersection (this problem is known as the Fan Raspaud Conjecture). A FR-triple is a set of 3 such perfect matchings. We show here how to derive a Fulkerson covering from two FR-triples.Moreover, we give a simple proof that the Fulkerson conjecture holds true for some classes of well known snarks.1991 Mathematics Subject Classification. 035 C.

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