Mathematics – Combinatorics
Scientific paper
2006-02-08
Mathematics
Combinatorics
42 pages, the second version is shorter, focusing on the bijection (with application to counting) and on the graph drawing alg
Scientific paper
This article focuses on a combinatorial structure specific to triangulated plane graphs with quadrangular outer face and no separating triangle, which are called irreducible triangulations. The structure has been introduced by Xin He under the name of regular edge-labelling and consists of two bipolar orientations that are transversal. For this reason, the terminology used here is that of transversal structures. The main results obtained in the article are a bijection between irreducible triangulations and ternary trees, and a straight-line drawing algorithm for irreducible triangulations. For a random irreducible triangulation with $n$ vertices, the grid size of the drawing is asymptotically with high probability $11n/27\times 11n/27$ up to an additive error of $\cO(\sqrt{n})$. In contrast, the best previously known algorithm for these triangulations only guarantees a grid size $(\lceil n/2\rceil -1)\times \lfloor n/2\rfloor$.
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