Two-connected signed graphs with maximum nullity at most two (Q2228486)

\textit{T. Zaslavsky} [Discrete Appl. Math. 4, 47--74 (1982; Zbl 0476.05080)] introduced signed graphs. Contracting an edge \(e\) with ends \(u\) and \(v\) in a graph \(G\) means deleting \(e\) and identifying the vertices \(u\) and \(v\). A signed graph is a pair \((G, \Sigma)\), where \(G = (V, E)\) is a graph (in which parallel edges are permitted, but loops are not) with \(V = \{1,\dots,n\}\) and \(\Sigma\subseteq E\). The edges in \(\Sigma\) are called odd and the other edges of \(E\) even. By \(S(G,\Sigma)\) denote the set of all symmetric \(n\times n\) matrices \(A = [a_{i,j}]\) with \(a_{i,j} < 0\) if \(i\) and \(j\) are adjacent and connected by only even edges, \(a_{i,j} > 0\) if \(i\) and \(j\) are adjacent and connected by only odd edges, \( a_{i,j}\in R\) if \(i\) and \(j\) are connected by both even and odd edges, \(a_{i,j} = 0\) if \(i = j\) and \(i\) and \(j\) are non-adjacent, and \(a_{i,i}\in R\) for all vertices \(i\). The parameters \(M(G,\Sigma)\) and \(\xi(G,\Sigma)\) of a signed graph \((G,\Sigma)\) are the largest nullity of any matrix \(A\in S(G,\Sigma)\) and the largest nullity of any matrix \(A\in S(G,\Sigma)\) that has the strong Arnold property, respectively. In an earlier paper, the authors [ibid. 439, No. 5, 1506--1529 (2013; Zbl 1282.05058)] gave a characterization of signed graphs \((G,\Sigma)\) with \(M(G,\Sigma)\leq 1\) and of signed graphs with \(\xi(G,\Sigma)\leq 1\). Here, they characterize the 2-connected signed graphs \((G,\Sigma)\) with \(M(G,\Sigma)\leq 2\) and the 2-connected signed graphs \((G,\Sigma)\) with \(\xi(G,\Sigma)\leq 2\). The concept of wide separation is the main frame of the article. This article is useful to researchers working on signed graphs and allied areas.