The Art Gallery Theorem for Polyominoes

We explore the art gallery problem for the special case that the domain (gallery) P is an mpolyomino, a polyform whose cells are m unit squares. We study the combinatorics of guarding polyominoes in terms of the parameter m, in contrast with the traditional parameter n, the number of vertices of P . In particular, we show that ⌊ 3 ⌋ point guards are always sufficient and sometimes necessary to cover an m-polyomino, possibly with holes. When m ≤ 3n 4 − 4, the sufficiency condition yields a strictly lower guard number than ⌊ 4 ⌋, given by the art gallery theorem for orthogonal polygons.