Transversal structures on triangulations: a combinatorial study and straight-line drawings
Eric Fusy

TL;DR
This paper introduces a combinatorial structure called transversal structures on irreducible triangulations, establishes a bijection with ternary trees, and presents a straight-line drawing algorithm with improved grid size bounds.
Contribution
It provides a new combinatorial characterization of irreducible triangulations and an efficient drawing algorithm with asymptotically optimal grid size.
Findings
Bijection between irreducible triangulations and ternary trees.
A straight-line drawing algorithm with asymptotically optimal grid size.
Improved grid size bounds compared to previous algorithms.
Abstract
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 vertices, the grid size of the drawing is asymptotically with high probability up to an additive error of . In contrast, the best previously known algorithm for these triangulations only guarantees a grid size…
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Advanced Graph Theory Research · Advanced Combinatorial Mathematics
