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

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 $(\lceil n/2\rceil -1)\times \lfloor n/2\rfloor$.Comment: 42 pages, the second version is shorter, focusing on the bijection (with application to counting) and on the graph drawing algorithm. The title has been slightly change

A Schnyder-type drawing algorithm for 5-connected triangulations

We define some Schnyder-type combinatorial structures on a class of planar triangulations of the pentagon which are closely related to 5-connected triangulations. The combinatorial structures have three incarnations defined in terms of orientations, corner-labelings, and woods respectively. The wood incarnation consists in 5 spanning trees crossing each other in an orderly fashion. Similarly as for Schnyder woods on triangulations, it induces, for each vertex, a partition of the inner triangles into face-connected regions (5~regions here). We show that the induced barycentric vertex-placement, where each vertex is at the barycenter of the 5 outer vertices with weights given by the number of faces in each region, yields a planar straight-line drawing.Comment: Appears in the Proceedings of the 31st International Symposium on Graph Drawing and Network Visualization (GD 2023

Grid Embedding of 4-Connected Plane Graphs

A straight line grid embedding of a plane graph G is a drawing of G such that the vertices are drawn at grid points and the edges are drawn as non-intersecting straight line segments. In this paper, we show that, if a 4-connected plane graph G has at least 4 vertices on its exterior face, then G can be embedded on a grid of size W \times H such that W + H \le n, W \le (n + 3)/2 and H \le 2(n - 1)/3, where n is the number of vertices of G. Such an embedding can be computed in linear time