Partitioning orthogonal polygons into at most 8-vertex pieces, with application to an art gallery theorem

TL;DR

This paper presents a new method to partition orthogonal polygons into smaller pieces, leading to a shorter proof of a guard placement theorem and extending it to more practical guard types, thus advancing art gallery problem solutions.

Contribution

It introduces a novel partitioning technique for orthogonal polygons into at most 8-vertex pieces, providing a shorter proof of a guard theorem and addressing practical guard constraints.

Findings

Partitioning orthogonal polygons into at most 8-vertex pieces.

Shorter proof of Aggarwal's guard theorem.

Positive answers to questions on combinatorial guards.

Abstract

We prove that every simply connected orthogonal polygon of $n$ vertices can be partitioned into $\left\lfloor\frac{3 n +4}{16}\right\rfloor$ (simply connected) orthogonal polygons of at most 8 vertices. It yields a new and shorter proof of the theorem of A. Aggarwal that $\left\lfloor\frac{3 n +4}{16}\right\rfloor$ mobile guards are sufficient to control the interior of an $n$-vertex orthogonal polygon. Moreover, we strengthen this result by requiring combinatorial guards (visibility is only required at the endpoints of patrols) and prohibiting intersecting patrols. This yields positive answers to two questions of O'Rourke. Our result is also a further example of the "metatheorem" that (orthogonal) art gallery theorems are based on partition theorems.