Multi-triangulations as complexes of star polygons

TL;DR

This paper introduces a novel perspective on k-triangulations by viewing them as complexes of star polygons, providing new proofs of their properties and opening new research directions.

Contribution

It presents a new interpretation of k-triangulations as complexes of star polygons, offering simpler proofs and new insights into their properties.

Findings

New interpretation of k-triangulations as star polygon complexes

Simplified proofs of fundamental properties

Exploration of new research directions

Abstract

Maximal $(k+1)$-crossing-free graphs on a planar point set in convex position, that is, $k$-triangulations, have received attention in recent literature, with motivation coming from several interpretations of them. We introduce a new way of looking at $k$-triangulations, namely as complexes of star polygons. With this tool we give new, direct, proofs of the fundamental properties of $k$-triangulations, as well as some new results. This interpretation also opens-up new avenues of research, that we briefly explore in the last section.