Skew rank decompositions

Abstract

We study simple graphs G of order n for which every n by n skew-symmetric matrix A with support in the edge set of G can be expressed as the sum of (rank A)/2, rank two skew-symmetric matrices with supports also in the edge set of G. We say that such graphs support skew rank decompositions (s.r.d.'s). These graphs generalize the bipartite graphs of order m by n that support rank decompositions of m by n matrices. The latter have recently been shown to be the chordal bipartite graphs, a class of bipartite graphs that arises when Gaussian elimination is to be performed with restricted fill-in. We also introduce a generalization of chordal bipartite graphs that arise in Gaussian elimination of skew-symmetric matrices. Finally, we examine s.r.d.'s that conform with a given sign pattern and obtain a graphical characterization of the sign patterns that support such signed s.r.d.'s.