Two-connected signed graphs with maximum nullity at most two

A signed graph is a pair $(G,Σ)$, where $G=(V,E)$ is a graph (in which parallel edges are permitted, but loops are not) with $V=\{1,\ldots,n\}$ and $Σ\subseteq E$. The edges in $Σ$ are called odd and the other edges of $E$ even. By $S(G,Σ)$ we 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, $a_{i,j}\in \mathbb{R}$ if $i$ and $j$ are connected by both even and odd edges, $a_{i,j}=0$ if $i\not=j$ and $i$ and $j$ are non-adjacent, and $a_{i,i} \in \mathbb{R}$ for all vertices $i$. The parameters $M(G,Σ)$ and $ξ(G,Σ)$ of a signed graph $(G,Σ)$ are the largest nullity of any matrix $A\in S(G,Σ)$ and the largest nullity of any matrix $A\in S(G,Σ)$ that has the Strong Arnold Hypothesis, respectively. In a previous paper, we gave a characterization of signed graphs $(G,Σ)$ with $M(G,Σ)\leq 1$ and of signed graphs with $ξ(G,Σ)\leq 1$. In this paper, we characterize the $2$-connected signed graphs $(G,Σ)$ with $M(G,Σ)\leq 2$ and the $2$-connected signed graphs $(G,Σ)$ with $ξ(G,Σ)\leq 2$.

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