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.
Article PDF
Similar content being viewed by others
Author information
Authors and Affiliations
Corresponding authors
Rights and permissions
About this article
Cite this article
Speckmann, B., Tóth, C. Allocating Vertex π-Guards in Simple Polygons via Pseudo-Triangulations. Discrete Comput Geom 33, 345–364 (2005). https://doi.org/10.1007/s00454-004-1091-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00454-004-1091-9