Art Galleries with Interior Walls

Consider an art gallery formed by a polygon on n vertices with m pairs of vertices joined by interior diagonals, the interior walls. Each interior wall has an arbitrarily placed, arbitrarily small doorway. We will show that the minimum number of guards that suffice to guard all art galleries with n vertices and m interior walls is min{b(2n−3)/3c, b(2n+m−2)/4c, b(2m+n)/3c}. If we restrict ourselves to galleries with convex rooms of size at least r, the answer improves to min {m, b(n + m)/rc}. The proofs lead to linear time guard placement algorithms in most cases. The original art gallery problem, posed by Klee and solved by Chvátal [6], is to find the smallest number of guards necessary to cover any simple polygon, the art gallery, not necessarily convex, on n vertices. Here a covering by g guards means that one can find g points in the interior of the polygon such that every point in the interior is covered by some guard, that is for each point in the interior the line segment between it and some guard does not intersect the polygon. The comb polygons in figure 1 show that bn/3c guards are sometimes necessary – if n is not divisible by 3 simply take a comb on 3bn/3c vertices and subdivide one or two of its edges. Chvátal also showed that bn/3c guards always suffice. For more information on the history of this problem and related problems, see [15] and [17]. $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ # # # # # # # # # # # # # # # # # # # # # # # $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ # # # # # # # # # # # # # # # # # # # # # # # Figure 1: Comb polygons Hutchinson [11] generalized the basic art gallery problem by allowing interior walls. Throughout this paper an art gallery (with interior walls) will be a simple polygon on n vertices with some pairs of vertices joined by non-intersecting interior diagonals, the interior walls. Also suppose that in the interior of each of the walls Typeset by AMS-TEX 1