The structure and labelled enumeration of K_{3,3}-subdivision-free projective-planar graphs

TL;DR

This paper characterizes a class of non-planar, K_{3,3}-subdivision-free graphs embeddable in the projective plane, providing a unique decomposition, enumeration methods, and bounds on edges.

Contribution

It introduces a unique decomposition of these graphs into a core K_5 with edges replaced by two-pole networks, and develops enumeration techniques for labelled graphs.

Findings

Graphs in the class have at most 3n-6 edges for n >= 6.

The class admits a unique decomposition involving K_5 and two-pole networks.

Enumeration methods for these graphs and their irreducible variants are described.

Abstract

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.