Hidden Mobile Guards in Simple Polygons

TL;DR

This paper investigates the existence of hidden mobile guards in various classes of simple polygons, providing comprehensive results on when such guards can or cannot be used for effective coverage.

Contribution

It offers a nearly complete characterization of the existence of hidden open and closed mobile guards in different polygon classes, revealing surprising limitations and possibilities.

Findings

Every monotone or starshaped polygon can be guarded with hidden open mobile guards.

Not all polygons admit hidden open edge or diagonal guards.

The paper provides nearly complete answers to existence questions for various guard types.

Abstract

We consider guarding classes of simple polygons using mobile guards (polygon edges and diagonals) under the constraint that no two guards may see each other. In contrast to most other art gallery problems, existence is the primary question: does a specific type of polygon admit some guard set? Types include simple polygons and the subclasses of orthogonal, monotone, and starshaped polygons. Additionally, guards may either exclude or include the endpoints (so-called open and closed guards). We provide a nearly complete set of answers to existence questions of open and closed edge, diagonal, and mobile guards in simple, orthogonal, monotone, and starshaped polygons, with some surprising results. For instance, every monotone or starshaped polygon can be guarded using hidden open mobile (edge or diagonal) guards, but not necessarily with hidden open edge or hidden open diagonal guards.