On the digraphs of sign solvable linear systems

Abstract

A linear system of equations Ax=b is sign solvable if it is solvable and both its solvability and the sign pattern of its solution vector x are uniquely determined by the sign patterns of A and b. It was proved in [L. Bassett, J. Maybee, J. Quirk, Economitrica 36 (1968) 544–563; R.A. Bruali, B.L. Shader, Matrices of Sign-Solvable Linear Systems, Cambridge University Press, Cambridge, 1995] that the sign solvability of a linear system Ax=b in standard form and with a square coefficient matrix A can be described in terms of the signed digraph S(A) of the matrix A with a specified vertex subset depending on the sign pattern of the vector b. In this paper we define the notion of W + -sign solvable signed digraphs which represent precisely the sign solvable linear systems in standard form. We give necessary and sufficient conditions for the underlying (unsigned) digraphs of W + -sign solvable signed digraphs in terms of forbidden subdigraphs. We also give several necessary and sufficient conditions for strongly connected W + -sign solvable signed digraphs.