Principally unimodular skew-symmetric matrices (Q1307441)

A square matrix is said to be principally unimodular (PU) if its every submatrix has determinant 0 or \(\pm 1\). Let \(A\) be a symmetric \((0, 1)\)-matrix, with zero diagonal. A PU-orientation of \(A\) is a skew-symmetric signing of \(A\) that is PU. Given a matrix \(A'\) that is a PU-orientation of \(A\), then the authors can construct all PU-orientations of \(A\). The idea of proof involves the decomposition of \(A\), and the fact that the PU-orientations of indecomposable matrices are unique up to negation and multiplication of certain rows/columns by \(-1\). In fact, the result can be considered as a generalization of \textit{P. Camion}'s earlier theorem [Cah. Cent. Étud. Rech. Opér. 5, 181-190 (1963; Zbl 0124.00901)] which states that if a \((0,1)\)-matrix can be signed to be totally unimodular then the signing is unique up to multiplying certain rows/columns by \(-1\). The later result is an important step in proving Tutte's classical theorem that gives an excluded minor characterization of totally unimodular matrices.