A characterization of projective-planar signed graphs (Q1772410)

A signed graph \(G^{\pm }\) is a graph \(G\) together with an assignment of a plus or minus sign on each edge. A simple cycle in a signed graph is positive or negative according to whether it contains an even or odd number of negative edges, respectively. An embedding of a signed graph in a surface is an embedding of the underlying graph such that a simple cycle is orientation reversing if and only if it is negative. This paper gives a characterization of which signed graphs embed in the projective plane. It is in terms of a related signed graph formed by considering the theta subgraphs of the given graph, namely: a loopless 2-connected signed graph \(G^{\pm }\) is projective planar if and only if \(G^{\pm }\) has a totally positive twist-free claw rotation system.