Guarding Polyominoes, Polycubes and Polyhypercubes

We consider variations of the original art gallery problem where the domain is a polyomino, a polycube, or a polyhypercube. Anm-polyomino is the connected union of m unit squares called pixels, an m-polycube is the connected union of m unit cubes called voxels, and an m-polyhypercube is the connected union of m unit hypercubes in a d dimensional Euclidean space. In this paper we generalize and unify the known results about guarding polyominoes and polycubes and obtain simpler proofs. We also obtain new art gallery theorems for guarding polyhypercubes.