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

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 $n$ 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 $(\lceil n/2\rceil -1)\times \lfloor n/2\rfloor$.