  We consider the class F of 2-connected non-planar K_{3,3}-subdivision-free
graphs that are embeddable in the projective plane. We show that these graphs
admit a unique decomposition as a graph K_5 (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 series-parallel networks. We also show that the number m of edges of
graphs in F with n vertices satisfies the bound m <=3n-6, for n >= 6.
