Matching structure of symmetric bipartite graphs and a generalization of Pólya's problem

A bipartite graph is said to be symmetric if it has symmetry of reflecting two vertex sets. This paper investigates matching structure of symmetric bipartite graphs. We first apply the Dulmage-Mendelsohn decomposition to a symmetric bipartite graph. The resulting components, which are matching-covered, turn out to have symmetry. We then decompose a matching-covered bipartite graph via an ear decomposition, which is a sequence of subgraphs obtained by adding an odd-length path repeatedly. We show that, if a matching-covered bipartite graph is symmetric, an ear decomposition can retain symmetry by adding no more than two paths. As an application of these decompositions to combinatorial matrix theory, we present a natural generalization of Pólya’s problem. We introduce the problem of deciding whether a rectangular {0, 1}-matrix has a signing that is totally sign-nonsingular or not, where a rectangular matrix is totally sign-nonsingular if the sign of the determinant of each submatrix with the entire row set is uniquely determined by the signs of the nonzero entries. We show that this problem can be solved in polynomial time with the aid of the matching structure of symmetric bipartite graphs. In addition, we provide a characterization of this problem in terms of excluded minors.