Perfect Graphs and Guarding Rectilinear Art Galleries

Abstract

In this paper we introduce a new method based on perfect graphs that can be used to prove bounds on the number of guards necessary to guard a rectilinear art gallery in terms of the number of vertices, area and perimeter, respectively. Using this method, we prove that a polyomino with perimeter l≥ 6 can be guarded by at most ⌊ l/6⌋ guards. Moreover, we give a new proof for the rectilinear art gallery theorem, stating that any rectilinear art gallery with n vertices can be guarded by at most ⌊n/4⌋ guards. Finally, we give a new proof that at most ⌊m+1/3⌋ guards are necessary to guard an m-polyomino, even if the polyomino contains holes. © 2014 Springer Science+Business Media New York.