On digraphs and forbidden configurations of strong sign nonsingular matrices Original Research Article

A square real matrix A called a strong sign nonsingular matrix (or “S2NS” matrix) if all matrices with the same sign pattern as A nonsingular and the inverses of these matrices all have the same sign pattern. A digraph which is the underlying digraph of the signed digraph of an S2NS matrix (with a negative main diagonal) is called an S2NS digraph. In [9], Thomassen gave a characterization of strongly connected S2NS digraphs in terms of the forbidden subdigraphs. In [2], Brualdi and Shader constructed minimal forbidden configurations for S2NS digraphs for the general cases where the digraphs considered are not necessarily strongly connected. They also proposed the problem about the existence of new minimal forbidden configurations other than those found in [2,9]. In this paper, we construct infinitely many new (basic) minimal forbidden configurations and thus obtain the answer this problem. We also obtain several necessary conditions for minimal forbidden configurations and give a generalization of Thomassenʹs Theorem.