On Polygons Excluding Point Sets

TL;DR

This paper investigates conditions under which a polygon formed from blue points can exclude all red points, establishing a polynomial bound on the number of interior blue points needed relative to red points.

Contribution

It introduces a polynomial bound on the number of interior blue points required to ensure a polygon excludes all red points, addressing a dual problem to previous inclusion results.

Findings

Existence of a polynomial bound K(l) for blue points to exclude red points

K(l) is polynomial in the number of red interior points l

Provides conditions for blue polygonizations that exclude red points

Abstract

By a polygonization of a finite point set $S$ in the plane we understand a simple polygon having $S$ as the set of its vertices. Let $B$ and $R$ be sets of blue and red points, respectively, in the plane such that $B\cup R$ is in general position, and the convex hull of $B$ contains $k$ interior blue points and $l$ interior red points. Hurtado et al. found sufficient conditions for the existence of a blue polygonization that encloses all red points. We consider the dual question of the existence of a blue polygonization that excludes all red points $R$. We show that there is a minimal number $K=K(l)$, which is polynomial in $l$, such that one can always find a blue polygonization excluding all red points, whenever $k\geq K$. Some other related problems are also considered.