On guarding polygons with holes

TL;DR

This paper proves a new theorem for guarding polygons with holes, specifically for vertex guards that dominate the visibility graph and cover the outer boundary, extending previous conjectures to a special case.

Contribution

It introduces a theorem for guarding polygons with holes using vertex guards that dominate the visibility graph and ensure coverage of the outer boundary, a novel extension.

Findings

Proves a theorem for vertex guards in polygons with holes

Guards form a dominating set in the visibility graph

Guards cover both vertices and outer boundary

Abstract

There is an old conjecture by Shermer \cite{sher} that in a polygon with $n$ vertices and $h$ holes, $\lfloor \dfrac{n+h}{3} \rfloor$ vertex guards are sufficient to guard the entire polygon. The conjecture is proved for $h=1$ by Shermer \cite{sher} and Aggarwal \cite{aga} seperately. In this paper, we prove a theorem similar to the Shermer's conjecture for a special case where the goal is to guard the vertices of the polygon (not the entire polygon) which is equivalent to finding a dominating set for the visibility graph of the polygon. Our proof also guarantees that the selected vertex guards also cover the entire outer boundary (outer perimeter of the polygon) as well.