Proceedings of the 23rd Annual Canadian Conference on Computational Geometry, CCCG 2011
We address the question: How many edge guards are needed to guard an orthogonal polyhedron of e edges, r of which are reflex? It was previously established  that e/12 are sometimes necessary and e/6 always suf- fice. In contrast to the closed edge guards used for these bounds, we introduce a new model, open edge guards (excluding the endpoints of the edge), which we argue are in some sense more natural in this context. After quantifying the relationship between closed and open edge guards, we improve the upper bound to show that, asymptotically, (11/72)e (open or closed) edge guards suffice, or, in terms of r, that (7/12)r suffice. Along the way, we establish tight bounds relating e and r for orthogonal polyhedra of any genus.
© the authors
Benbernou, Nadia M.; Demaine, Erik D.; Demaine, Martin L.; Kurdia, Anastasia; O'Rourke, Joseph; Toussaint, Godfried; Urrutia, Jorge; and Viglietta, Giovanni, "Edge-Guarding Orthogonal Polyhedra" (2011). Computer Science: Faculty Publications, Smith College, Northampton, MA.