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

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 ∈ P if the line segment ℓ 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 b covers p 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 ϵ > 0 such that any point in P at distance at most ϵ 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 ⌊ n / 8 ⌋ light sources placed at vertices of P , and that all of the points in P can always be covered with at most ⌊ n / 12 ⌋ vertex beacons. Both bounds are tight.