The S\(^2\)NS digraphs and the cycle linear system of a digraph (Q1862950)

Let \(A\) be a square real matrix of order \(n\), and \(Q(A)\) be a set of all real matrices with the same sign pattern as \(A.\) Then \(A\) is said to be a strong sign nonsingular matrix, an \(\text{S}^{2}\text{NS}\) matrix, if each matrix in \(Q(A) \) is nonsingular and the inverses of all matrices in \(Q(A)\) have the same sign pattern. The signed digraph \(\text{SD}(A)\) of \(A\) is the digraph with the vertex set \(V=\{1,2,\dots,n\},\) and the arc set \(A=\{(i,j)\mid a_{ij}\neq 0\},\) where the sign of \(a_{ij}\) is attached to the arc \((i,j).\) A signed digraph \(S\) is called an \(\text{S}^{2}\text{NS}\) digraph if the arcs of \(S\) can be suitably assigned signs so that the resulting digraph is the \(\text{SD}(A)\) digraph of an \( \text{S}^{2}\text{NS}\) matrix \(A.\) A signed digraph which is not an \(\text{S}^{2}\text{NS}\) digraph is called a forbidden configuration. A graph theoretical characterization of \(\text{S}^{2}\text{NS}\) digraphs is given as well as an infinite class of the minimal forbidden configurations.