Discrete & Computational Geometry

, Volume 33, Issue 2, pp 345–364

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


    • Department of Mathematics and Computer Science, TU Eindhoven, P.O. Box 513, Eindhoven 5600 MB
    • Department of Computer Science, University of California at Santa Barbara, Santa Barbara, CA 93106

DOI: 10.1007/s00454-004-1091-9

Cite this article as:
Speckmann, B. & Tóth, C. Discrete Comput Geom (2005) 33: 345. doi:10.1007/s00454-004-1091-9


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.

Copyright information

© Springer Science+Business Media, Inc. 2004