Modem illumination of monotone polygons

Abstract 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 2 k + 3 ⌉ k-modems. In addition, we exhibit examples of monotone polygons requiring at least ⌈ n − 2 2 k + 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 2 k + 4 ⌉ k-modems, while for odd k ≥ 3 , ⌈ n − 2 2 k + 6 ⌉ k-modems are always sufficient. Further, by presenting according examples of monotone orthogonal polygons, we show that both bounds are tight.