A Canonical m-Atomic Decomposition of Bipartite Graphs via a Grid Model

Abstract

We study finite, connected, simple bipartite graphs in a grid model, in which a graph is drawn as a rectangular array and its structure is read off from empty subrectangles, called holes. In this model we attach to every brick a numerical invariant, its characteristic m, the difference between the number of rows and the largest proper independent set. A brick is excessive if m > 0. Our main results concern this invariant. We determine the characteristic of a disconnected excessive brick from those of its components, showing that m = min_i min{m_i, imb(W_i)} while the imbalance is additive; and we prove that an m-excessive brick is m-extendable, that is, every matching of size m extends to a maximum matching. Since Plummer's notion of n-extendability is defined only for graphs carrying a perfect matching, and our proof nowhere uses balance, the characteristic extends that notion canonically to unbalanced bipartite graphs. Using the characteristic we partition bipartite graphs into eleven structural classes. The underlying decomposition into atomic blocks is the classical decomposition into elementary components, and the description of the maximum proper independent sets by ideals of the block poset is likewise classical; the paper states precisely which results are classical and are not claimed here. What the grid model adds is a single geometric framework in which holes, characteristics and the block triangular form are read off from one picture.