The Art Gallery Theorem for Polyominoes

Abstract

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 ⌊ m+1 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 ⌊ n 4 ⌋, given by the art gallery theorem for orthogonal polygons.

Art Gallery Problem with Rook and Queen Vision

“…Our first pair of results are analogous to the result that gives the minimum number of guards needed to see all of an n-tile polyomino [5]. The difference is that our guards are rooks or queens rather than points with standard vision.…”

“…The question is similar to the famous art gallery problem of Chvátal, in which b n 3 c guards are sufficient to see all points in an n-vertex polygon, and some polygons need this many guards [8]. The paper first introducing the art gallery problem for polyominoes showss that b nþ1 3 c guards are sufficient and sometimes necessary to see all points in an n-tile polyomino [5], and there are similar bounds for higherdimensional polycubes [24]. Different notions of vision are possible in art gallery problems.…”

Guarding Orthogonal Art Galleries Using Sliding Cameras: Algorithmic and Hardness Results

“…Moreover, if all line segments in S are vertical or all are horizontal, then they cannot collectively guard the outer rectangle entirely. 2 In order to guard P entirely, we add one more orthogonal line segment C as follows: if all line segments in S are vertical (resp., horizontal), then C is the maximal horizontal (resp., the maximal vertical) line segment inside P that aligns the upper edge (resp., the right edge) of the smaller rectangle of P ; see the line segment e (resp., e ) in Figure 6. If the line segments in S are a combination of vertical and horizontal line segments, then C can be either e or e .…”

“…The art gallery problem is well known in computational geometry, where the objective is to cover a geometric shape (e.g., a polygon) with the union of the visibility regions of a set of point guards while minimizing the number of guards. The problem's multiple variants have been examined extensively (e.g., see [1,15,17]) and can be classified based on the type of guards (e.g., points or line segments), the type of visibility model, and the geometric shape (e.g., simple polygons, orthogonal polygons [6], or polyominoes [2]).…”

A (7/2)-Approximation Algorithm for Guarding Orthogonal Art Galleries with Sliding Cameras

“…Moreover, the problem was also shown to be NP-hard for orthogonal polygons [14]. Since then, the problem and its many variants have been studied extensively for different types of polygons (e.g., orthogonal polygons [14] and polyominoes [1]), different types of guards (e.g., points and line segments) and different visibility types. See the surveys by O'Rourke [13] or Urrutia [15] for a detailed history of the art gallery problem.…”