Structure and labelled enumeration of K3,3-subdivision-free projective-planar graphs

We consider the class F of 2-connected non-planar K3,3-subdivision-free graphs that are embeddable in the projective plane. We show that these graphs admit a unique decomposition as a graph K5 (the core) where the edges are replaced by two-pole networks constructed from 2-connected planar graphs. A method to enumerate these graphs in the labelled case is described. Moreover, we enumerate the homeomorphically irreducible graphs in F and homeomorphically irreducible 2-connected planar graphs. Particular use is made of two-pole (directed) seriesparallel networks. We also show that the number m of edges of graphs in F satisfies the bound m ≤ 3n − 6, for n ≥ 6 vertices.