Edge-guarding Orthogonal Polyhedra

Abstract

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 [3] that e/12 are sometimes necessary and e/6 always suffice. 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 ofr, that (7/12)r suffice. Along the way, we establish tight bounds relating e and r for orthogonal polyhedra of any genus.