Displaying blocking pairs in signed graphs

A signed graph is a pair (G,Σ) where G is a graph and Σ is a subset of the edges of G. A circuit of G is even (resp. odd) if it contains an even (resp. odd) number of edges of Σ. A blocking pair of (G,Σ) is a pair of vertices s, t such that every odd circuit intersects at least one of s or t. In this paper, we characterize when the blocking pairs of a signed graph can be represented by 2-cuts in an auxiliary graph. We discuss the relevance of this result to the problem of recognizing even cycle matroids and to the problem of characterizing signed graphs with no odd-K5 minor.