Edge guarding polyhedral terrains

Art Gallery and Illumination Problems

“…A set S of edges of T guards it if every triangle of T has a vertex that is the end-vertex of an edge in S. A set H of vertices of T guards it if every face of T has a vertex in H. We will prove the following results, proved by Everett and Rivera-Campo [53], and Bose, Shermer, Toussaint and Zhu [16] respectively: Theorem 7.1.1 Any triangulation of a set of n points can be guarded with at most n 3 edges. Theorem 7.1.2 Any triangulation of a point set can be guarded with at most n 2 vertices.…”

Improved Bounds for Guarding Plane Graphs with Edges

“…Such a set is said to guard G. We improve the known upper bounds on the number of edges required to guard any n-vertex embedded planar graph G:1. We present a simple inductive proof for a theorem of Everett and Rivera-Campo (1997) that G can be guarded with at most 2n 5 edges, then extend this approach with a deeper analysis to yield an improved bound of 3n 8 edges for any plane graph. 2.…”

Lower Bound of Face Guards of Polyhedral Terrains

“…The definitions of polyhedral terrains and visibility are mostly from landmark papers on guarding polyhedral terrains [3], [5]. A polyhedral terrain is a polyhedral surface in three dimensions such that its intersection with any vertical line is either a point or empty.…”

“…In Table 1, upper bounds of vertex and edge guards were firstly proved in 1997 [3], [5]. However, these bounds are based on the four color theorem, and for this reason, there seemed to be no practical efficient algorithms achieving these bounds.…”