Transversal Structures on Triangulations, with Application to Straight-Line Drawing

Abstract

Abstract. We define and investigate a structure called transversal edgepartition related to triangulations without non empty triangles, which is equivalent to the regular edge labeling discovered by Kant and He. We study other properties of this structure and show that it gives rise to a new straight-line drawing algorithm for triangulations without non empty triangles, and more generally for 4-connected plane graphs with at least 4 border vertices. Taking uniformly at random such a triangulation with 4 border vertices and n vertices, the size of the grid is almost surely 11 27 n × 11 27 n up to fluctuations of order √ n, and the halfperimeter is bounded by n − 1. The best previously known algorithms for straight-line drawing of such triangulations only guaranteed a grid of size ( n/2 − 1) × n/2 . Hence, in comparison, the grid-size of our algorithm is reduced by a factor 5 27, which can be explained thanks to a new bijection between ternary trees and triangulations of the 4-gon without non empty triangles.

Straight-Line Drawing of Quadrangulations

“…A transversal structure is an orientation and partition of the inner edges of G into red and blue edges, satisfying the following conditions (see Figure 1 Transversal structures have been defined in [6,9] in the case where all inner faces are triangles. However, as we will see, other plane graphs can be endowed with a…”

“…Let G be a partially triangulated quadrangulation endowed with a transversal structure. Then, it is easily shown (adapting the proof of [6,Prop.1]) that the red edges of G form a bipolar orientation with source S and sink N ; and the blue edges form a bipolar orientation with source W and sink E. (We recall that a bipolar orientation is an acyclic orientation with a unique vertex having only outgoing edges, called source, and a unique vertex having only ingoing edges, called sink. )…”

“…Essentially there are two classes of linear time algorithms: a) purely iterative algorithms based on a specific order of treatment of vertices, the drawing being globally updated at each step [5,8,10]; b) oneshot algorithms, where the plane graph is first endowed with a combinatorial structure (e.g. Schnyder woods for triangulations), which is used to compute the coordinates of vertices, usually using some face-counting operations [11,2,6]. The algorithms of the second class have the advantage that the coordinates of vertices are computed independently, so that they are easier to implement and to perform on a piece of paper.…”

“…We use the quadri-partition to triangulate partially Q into a plane graph G endowed with a so-called transversal structure, i.e., a bicoloration and orientation of the inner edges of G with specific local conditions. Transversal structures were originally only defined for triangulations of a 4-gon [6,9]. The definition is easily extended to partially triangulated quadrangulations in Section 2.…”

“…Then, the transversal structure is used to draw G (hence, also Q) with face-counting operations. To do this, we extend the algorithm of [6], which draws triangulations of the 4-gon, to partially triangulated quadrangulations. The obtained drawing of Q verifies W + H = n − 1 − Δ, where n is the number of vertices and Δ is an explicit parameter of Q, see Theorem 1.…”