Signed graphs whose signed Colin de Verdière parameter is 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 , ? , n } and Σ ? E . The edges in Σ are called odd and the other edges even. By S ( G , Σ ) we denote the set of all real symmetric n × n matrices A = a i , j with a i , j < 0 if i and j are adjacent and all edges between i and j are even, a i , j 0 if i and j are adjacent and all edges between i and j are odd, and a i , j = 0 if i ? j and i and j are non-adjacent. The parameter ? ( G , Σ ) of a signed graph ( G , Σ ) is the largest nullity of any positive semidefinite matrix A ? S ( G , Σ ) that has the Strong Arnold Property. By K 3 = we denote the signed graph obtained from ( K 3 , ? ) by adding to each even edge an odd edge in parallel. In this paper, we prove that a signed graph ( G , Σ ) has ? ( G , Σ ) ? 2 if and only if ( G , Σ ) has no minor isomorphic to ( K 4 , E ( K 4 ) ) or K 3 = .