A Generalization of Tutte's Characterization of Totally Unimodular Matrices

An integral square matrix A is called principally unimodular (PU if every nonsingular principal submatrix is unimodular (that is, has determinant \1). Principal unimodularity was originally studied with regard to skew-symmetric matrices; see [2, 4, 5]; here we consider symmetric matrices. Our main theorem is a generalization of Tutte's excluded minor characterization of totally unimodular matrices; the generalization arises in the following way: a matrix B is totally unimodular if and only if the matrix ( 0 B T B 0 ) is PU. Before stating the main theorem we need to introduce some terminology. A signing of a symmetric (0, 1)-matrix A=(aij) is a symmetric (0, \1)matrix, say A$=(a$ij), such that aij=|a$ij |, for all i, j. We are concerned with the symmetric (0, 1)-matrices that admit a signing which is PU; such a signing is called a PU-signing. Let A be a V by V matrix, where V is a finite set. An isomorphism of A is a matrix obtained from A by a relabeling of its ground set V. (Note that isomorphisms freely allow simultaneous row column exchanges.) We denote by A[X] the principal submatrix of A induced by the set X V. For a set X V such that A[X] is nonsingular, define matrices P, Q, R, S, such that P=A[X] and A=( P R Q S ). Then define