A note on the lower bound of edge guards of polyhedral terrains

Abstract

Bose et al. [P. Bose et al., Guarding polyhedral terrains, Comput. Geom., Theory Appl. 6 (1997), pp. 173–185.] proved that ⌊ (4n−4)/13 ⌋ edge guards are sometimes necessary to guard an n-vertex polyhedral terrain. Subsequently, Kaučič et al. [B. Kaučič, B. Žalik, and F. Novak, On the lower bound of edge guards of polyhedral terrains, Int. J. Comput. Math. 80 (2003), pp. 811–814.] claimed to find an inconsistency in the proof and used Bose et al. ’s proof technique to prove a weaker lower bound of ⌊ (2n−4)/7 ⌋ edge-guards. They declared that a proof of the original lower bound of ⌊ (4n−4)/13 ⌋ remains an open issue. The purpose of this note is simply to point out that the issue is not open and that Bose et al. ’s original proof is correct. We present the original proof of ⌊ (4n−4)/13 ⌋ at a level of detail to hopefully remove any misunderstanding of the result.

Keywords:

• computational geometry
• graph theory
• edge guards
• planar graphs
• polyhedral terrains

2000 AMS Subject Classification :

• 32Bxx
• 32B25
• 05Cxx
• 05C15

Acknowledgements

The author would like to thank the referees for their help in improving the presentation of this paper. This research is supported in part by NSERC.

Notes

By V(G) we mean the vertex set of graph G.

A planar graph is outerplanar provided that one face of the graph is adjacent to all the vertices. An outerplanar graph is maximal if the graph is no longer outerplanar with the addition of a single edge. The face adjacent to all the vertices is usually referred to as the outerface.