Tight Bounds for Illuminating and Covering of Orthotrees with Vertex Lights and Vertex Beacons

Abstract

We consider two variants of the Art Gallery Problem: illuminating orthotrees with a minimum set of vertex lights, and covering orthotrees with a minimum set of vertex beacons. An orthotree P is a simply connected orthogonal polyhedron that is the union of a set S of cuboids glued face to face such that the graph whose vertices are the cuboids of S, two of which are adjacent if they share a common face, is a tree. A point p illuminates a point $$q \in P$$ if the line segment $$\ell$$ joining them is contained in P. A beacon b is a point in P that pulls other points in P towards itself similarly to the way a magnet attracts ferrous particles. We say that a beacon bcoversp if when b starts pulling p, p does not get stuck at a point of P before it reaches b. This happens, for instance if p reaches a point $$p'$$ such that there is an $$\epsilon >0$$ such that any point in P at distance at most $$\epsilon$$ from $$p'$$ is farther away from $$p'$$ than q (there is another pathological case that we will not detail in this abstract). In this paper we prove that any orthotree P with n vertices can be illuminated using at most $$\lfloor n/8 \rfloor$$ light sources placed at vertices of P, and that all of the points in P can always be covered with at most $$\lfloor n/12 \rfloor$$ vertex beacons. Both bounds are tight.