The projective-planar signed graphs (Q2366021)

A signature on a graph is an assignment of a plus or minus sign on each edge. A polygon in a signed graph is plus or minus depending on whether the number of edges with a minus sign is even or odd. An embedding of a signed graph embeds the underlying graph on some surface such that the minus polygons are precisely the orientation-reversing ones. In particular, if a signed graph is embedded in the projective plane, then the plus polygons are precisely the contractible ones. The author describes which signed graphs embed in the real projective plane by giving a list of six forbidden link minors (a link minor is a slight variation of a minor in which only links may be contracted). The six minors are based on the two Kuratowski graphs, \(K_ 5\) and \(K_{3,3}\). The author also gives a similar characterization in terms of eight forbidden topological subgraphs.