On Polygons Excluding Point Sets
Radoslav Fulek, Bal\'azs Keszegh, Filip Mori\'c, Igor Uljarevi\'c

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 in the plane we understand a simple polygon having as the set of its vertices. Let and be sets of blue and red points, respectively, in the plane such that is in general position, and the convex hull of contains interior blue points and 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 . We show that there is a minimal number , which is polynomial in , such that one can always find a blue polygonization excluding all red points, whenever . Some other related problems are also considered.
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Taxonomy
TopicsComputational Geometry and Mesh Generation · graph theory and CDMA systems · Advanced Graph Theory Research
