Counting Labelled Projective-planar Graphs without a K 3,3 -subdivision *

We consider the class F of labelled 2-connected non-planar graphs without a K3,3-subdivision that are embeddable in the projective plane. A method of enumerating these graphs is described. We also enumerate the homeomorphically irreducible graphs in F and homeomorphically irreducible 2-connected planar graphs. The methods are based on the projective-planarity characterization for graphs without a K3,3-subdivision [A. Gagarin and W. Kocay, “Embedding graphs containing K5-subdivisions”, Ars Combinatoria, 64 (2002), 33-49], and enumeration techniques involving the substitution of (0, 1)-networks for graph edges [T.R.S. Walsh, ”Counting labelled three-connected and homeomorphically irreducible two-connected graphs”, J. Comb. Theory Ser. B, 32 (1982), 1-11]. Particular use is made of twopole directed series-parallel networks. We also show that the number m of edges of graphs in F satisfies m ≤ 3n − 6, for n ≥ 6 vertices.