Some Results on Open Edge and Open Mobile Guarding of Polygons and Triangulations

TL;DR

This paper investigates the minimum number of open edge and open mobile guards needed to cover various classes of polygons and triangulation graphs, extending the classical Art Gallery problem with new guard types.

Contribution

It introduces and analyzes the concepts of open edge and open mobile guards, providing bounds and conditions for their sufficiency and necessity in polygon guarding.

Findings

Determines guard number bounds for certain polygon classes.

Establishes relationships between open guards and triangulation graphs.

Provides new insights into guard placement strategies.

Abstract

This paper focuses on a variation of the Art Gallery problem that considers open edge guards and open mobile guards. A mobile guard can be placed on edges and diagonals of a polygon, and the "open" prefix means that the endpoints of such edge or diagonal are not taken into account for visibility purposes. This paper studies the number of guards that are sufficient and sometimes necessary to guard some classes of simple polygons for both open edge and open mobile guards. This problem is also considered for planar triangulation graphs using open edge guards.