Allocating Vertex π-Guards in Simple Polygons via Pseudo-Triangulations

Authors:
Bettina Speckmann;Csaba D. Tóth
Affiliations:
Department of Mathematics and Computer Science, TU Eindhoven, P.O. Box 513, Eindhoven 5600 MB, The Netherlands;Department of Computer Science, University of California at Santa Barbara, Santa Barbara, CA 93106, USA
Venue:
Discrete & Computational Geometry
Year:
2005

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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 \lfloor n/2 \rfloor 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 \lfloor (2n – k)/3 \rfloor edge-aligned vertexπ-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.