Characterization of even directed graphs
Journal of Combinatorial Theory Series B
Pfaffian orientations 0-1 permanents, and even cycles in directed graphs
Discrete Applied Mathematics - Combinatorics and complexity
A new proof of a characterisation of Pfaffian bipartite graphs
Journal of Combinatorial Theory Series B
Upper degree-constrained partial orientations
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Computing the inertia from sign patterns
Mathematical Programming: Series A and B
Journal of Graph Theory
Solving linear programs from sign patterns
Mathematical Programming: Series A and B
Matrices and Matroids for Systems Analysis
Matrices and Matroids for Systems Analysis
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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 Polya'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.