Skew rank decompositions

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 …