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

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

2004-06-08

arxiv.org/abs/math/0406140v5

Pure Math. Appl. 16 (2005), no. 3, pp. 267-286

Mathematics

Combinatorics

Revised version. 20 pages. To appear in Pure Mathematics and Applications - Algebra and Theoretical Computer Science

Scientific paper

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.

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