Universal Guard Problems

TL;DR

This paper investigates the Universal Guard Problem, providing bounds and algorithms for selecting minimal guard sets that can guard any polygon within a class, advancing understanding of universal guarding strategies.

Contribution

It offers new upper and lower bounds on the number of universal guards needed and presents algorithms for constructing such guard sets.

Findings

Upper bounds on universal guard set size

Lower bounds on universal guard set size

Algorithms for constructing guard sets

Abstract

We provide a spectrum of results for the Universal Guard Problem, in which one is to obtain a small set of points ("guards") that are "universal" in their ability to guard any of a set of possible polygonal domains in the plane. We give upper and lower bounds on the number of universal guards that are always sufficient to guard all polygons having a given set of n vertices, or to guard all polygons in a given set of k polygons on an n-point vertex set. Our upper bound proofs include algorithms to construct universal guard sets of the respective cardinalities.