Which point sets admit a k-angulation?

TL;DR

This paper investigates the conditions under which a set of points in the plane admits a k-angulation, revealing that for sufficiently large sets, the only obstructions are those derived from Euler's formula.

Contribution

It characterizes when point sets admit k-angulations, showing that for large enough sets, only Euler's formula constraints prevent such embeddings.

Findings

Sets with at least 2k^2 points admit k-angulations unless obstructed by Euler's formula.

The only obstructions for large point sets are those implied by Euler's formula.

Provides conditions for the existence of k-angulations in plane point sets.

Abstract

For k >= 3, a k-angulation is a 2-connected plane graph in which every internal face is a k-gon. We say that a point set P admits a plane graph G if there is a straight-line drawing of G that maps V(G) onto P and has the same facial cycles and outer face as G. We investigate the conditions under which a point set P admits a k-angulation and find that, for sets containing at least 2k^2 points, the only obstructions are those that follow from Euler's formula.