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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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.

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