Vertex polygons

Fast Vertex Guarding for Polygons

For a polygon P with n vertices, the vertex guarding problem asks for the minimum subset G of P ’s vertices such that every point in P is seen by at least one point in G. This problem is NP-complete and APX-hard. The first approximation algorithm (Ghosh, 1987) involves decomposing P into O ( n ) cells that are equivalence classes for visibility from the vertices of P . This discretized problem ...

Vertex Guarding in Weak Visibility Polygons

The art gallery problem enquires about the least number of guards that are sufficient to ensure that an art gallery, represented by a polygon P , is fully guarded. In 1998, the problems of finding the minimum number of point guards, vertex guards, and edge guards required to guard P were shown to be APX-hard by Eidenbenz, Widmayer and Stamm. In 1987, Ghosh presented approximation algorithms for...

Polygons Whose Vertex Triangles Have Equal Area

Slow relaxation in weakly open vertex-splitting rational polygons

The problem of splitting effects by vertex angles is discussed for nonintegrable rational polygonal billiards. A statistical analysis of the decay dynamics in weakly open polygons is given through the orbit survival probability. Two distinct channels for the late-time relaxation of type tδ are established. The primary channel, associated with the universal relaxation of ”regular” orbits, with δ...

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

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 bo...