On Pseudo-Convex Partitions of a Planar Point Set

TL;DR

This paper improves bounds on the pseudo-convex partition number for planar point sets by establishing that =3, leading to a tighter upper bound and answering a previously posed open question.

Contribution

It proves that =3, which refines the upper bound on the pseudo-convex partition number for planar point sets.

Findings

=3, the exact pseudo-convex partition number for 13 points

Improved upper bound on (n) to n/13

Answers an open question from prior research

Abstract

Aichholzer et al. [{\it Graphs and Combinatorics}, Vol. 23, 481-507, 2007] introduced the notion of pseudo-convex partitioning of planar point sets and proved that the pseudo-convex partition number $\psi(n)$ satisfies, $\frac{3}{4}\lfloor\frac{n}{4}\rfloor\leq \psi(n)\leq\lceil\frac{n}{4}\rceil$. In this paper we prove that $\psi(13)=3$, which immediately improves the upper bound on $\psi(n)$ to $\lceil\frac{3n}{13}\rceil$, thus answering a question posed by Aichholzer et al. in the same paper.