Watched guards in art galleries

Abstract

The art gallery problem asks how many guards are sufficient to see every point of the interior of a polygon. A set of guards is called watched if each guard itself is seen by at least one of its colleagues. In 1994, Hernandez-Penalver wrote that \( \lfloor {\frac{{2n}} {5}} \rfloor \)watched guards always suffice to guard any polygon with n vertices. However in 2001, Michael and Pinciu, and independently Żylinski, presented a class of polygons that required more than \( \lfloor {\frac{{2n}} {5}} \rfloor \)watched guards – which disproved the Hernandez-Penalver’s result – and they established a new tight bound for watched guards: \( \lfloor {\frac{{3n - 1}} {7}} \rfloor \). Combinatorial bounds for watched guards in orthogonal polygons were independently given by Hernandez-Penalver , and by Michael and Pinciu, who proved the \( \lfloor {\frac{{n}} {3}} \rfloor \) -bound to be tight. In this paper, tight bounds for polygons of miscellaneous shapes are presented: \( \lfloor {\frac{{2n}} {5}} \rfloor \) watched guards for monotone and spiral polygons, and \( \lfloor {\frac{{3n - 1}} {7}} \rfloor \) vertex watched guards for star-shaped polygons.