Modem Illumination of Monotone Polygons

We study a generalization of the classical problem of 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 necessary to illuminate monotone and monotone orthogonal polygons. We show that every monotone polygon on $n$ vertices can be illuminated with $leftlceil n2k right $k$-modems and exhibit examples of monotone polygons requiring $leftlceil n2k+2 right $k$-modems. For monotone orthogonal polygons, we show that every such polygon on $n$ vertices can be illuminated with $leftlceil n2k+4 right $k$-modems and give examples which require $leftlceil n2k+4 right $k$-modems for $k$ even and $leftlceil n2k+6 right for $k$ odd.