Allocating vertex π-guards in simple polygons via pseudo-triangulations

Abstract

We use the concept of pointed pseudo-triangulations to establish new upper and lower bounds on a well known problem from the area of art galleries: What is the worst case optimal number of vertex π-guards that collectively monitor a simple polygon with n vertices? Our results are as follows: 1. Any simple polygon with n vertices can be monitored by at most ⌊n/2⌋ general vertex π-guards. This bound is tight up to an additive constant of 1. 2. Any simple polygon with n vertices, k of which are convex, can be monitored by at most ⌊(2n - k)/3⌊ edge-aligned vertex π-guards. This is the first non-trivial upper bound for this problem and it is tight for the worst case families of polygons known so far.