Modern illumination of monotone polygons

We study a generalization of the classical problem of the illumination of polygons. Instead of modeling a light source we model a wireless device whose radio signal can penetrate a given number k of walls. We call these objects k-modems and study the minimum number of k-modems sufficient and sometimes necessary to illuminate monotone and monotone orthogonal polygons. We show that every monotone polygon with n vertices can be illuminated with [n-2/2k+3] k-modems. In addition, we exhibit examples of monotone polygons requiring at least [n-2/2k+3] k-modems to be illuminated. For monotone orthogonal polygons with n vertices we show that for k = 1 and for even k, every such polygon can be illuminated with [n-2/2k+4] k-modems, while for odd k >= 3, [n-2/2k+6] k-modems are always sufficient. Further, by presenting according examples of monotone orthogonal polygons, we show that both bounds are tight. (C) 2017 Elsevier B.V. All rights reserved.