Straight Line Triangle Representations

TL;DR

This paper characterizes graphs that can be represented as straight line drawings with all faces as triangles, using flat angle assignments and contact systems of pseudosegments, and discusses their stretchability and recognition challenges.

Contribution

It provides new characterizations of graphs with straight line triangle representations based on flat angle assignments and pseudosegment contact systems, and explores stretchability via harmonic functions.

Findings

Characterization based on flat angle assignments and pseudosegment contact systems.

Stretchability of pseudosegment contact systems using discrete harmonic functions.

Open problem: deciding polynomial tractability of recognizing such graphs.

Abstract

A straight line triangle representation (SLTR) of a planar graph is a straight line drawing such that all the faces including the outer face have triangular shape. Such a drawing can be viewed as a tiling of a triangle using triangles with the input graph as skeletal structure. In this paper we present a characterization of graphs that have an SLTR. The characterization is based on flat angle assignments, i.e., selections of angles of the graph that have size~$\pi$ in the representation. We also provide a second characterization in terms of contact systems of pseudosegments. With the aid of discrete harmonic functions we show that contact systems of pseudosegments that respect certain conditions are stretchable. The stretching procedure is then used to get straight line triangle representations. Since the discrete harmonic function approach is quite flexible it allows further applications, we mention some of them. The drawback of the characterization of SLTRs is that we are not able to effectively check whether a given graph admits a flat angle assignment that fulfills the conditions. Hence it is still open to decide whether the recognition of graphs that admit straight line triangle representation is polynomially tractable.