Principally Unimodular Skew-Symmetric Matrices

principally unimodular (PU)

if every principal submatrix has determinant 0 or ±1. Let A be a symmetric (0, 1)-matrix, with a zero diagonal. A PU-orientation of A is a skew-symmetric signing of A that is PU. If A′ is a PU-orientation of A, then, by a certain decomposition of A, we can construct every PU-orientation of A from A′. This construction is based on the fact that the PU-orientations of indecomposable matrices are unique up to negation and multiplication of certain rows and corresponding columns by −1. This generalizes the well-known result of Camion, that if a (0, 1)-matrix can be signed to be totally unimodular then the signing is unique up to multiplying certain rows and columns by −1. Camion's result is an easy but crucial step in proving Tutte's famous excluded minor characterization of totally unimodular matrices.

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Received: May 17, 1996

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Bouchet, A., Cunningham, W. & Geelen, J. Principally Unimodular Skew-Symmetric Matrices. Combinatorica 18, 461–486 (1998).

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  • AMS Subject Classification (1991) Classes:  05C50, 05C85, 05B35