One-sided terrain guarding and chordal graphs

Abstract

The Terrain Guarding problem, a variant of the famous Art Gallery problem, has garnered significant attention over the last two decades in Computational Geometry from the viewpoint of complexity and approximability. Both the continuous and discrete versions of the problem were shown to be NP-Hard in King and Krohn (2010) and to admit a PTAS (Krohn et al., 2014; Friedrichs et al., 2016). The biggest unsolved question regarding this problem is if it is fixed-parameter tractable with respect to the size of the guard set. In this paper, we present two theorems that establish a relationship between a restricted case of the ONE-SIDED TERRAIN GUARDING problem and the Clique Cover problem in chordal graphs. These theorems were proved in Katz and Roisman (2008) for a special class of terrains called orthogonal terrains and were used to present a FPT algorithm with respect to the parameter that we require for DISCRETE ORTHOGONAL TERRAIN GUARDING in Ashok et al. (2018). We hope that the results obtained in this paper can, in future work, be used to produce such an algorithm for Discrete Terrain Guarding.