Coloring Variations of the Art Gallery Problem

The art gallery problem [2] asks for the smallest possible size of a point set S (the guards) to completely guard the interior of a simple polygon P (the art gallery). This thesis treats variations of the original problem that arise when we introduce a coloring of the guards. Rather than asking for the minimum number of guards we ask for the minimum number of colors. In [4] L. H. Erickson and S. M. LaValle introduced the following coloring: Two guards should be given different colors if their visibility regions intersect. What is the minimum number of colors required to color any guard set of P? We call this number the strong chromatic guard number of P, χsG(P). In the first part of the thesis (chapter 3) we show that for all positive integers n: there exists a polygon Pn with n vertices and χsG(Pn) = Ω(n) and an orthogonal polygon Pn with n vertices and χsG(Pn) = Ω( √ n). In the main part of the thesis (chapters 4 and 5) we show that when slightly changing the problem definition by using a conflict-free coloring of the guards, we can get polylogarithmic upper bounds for the conflictfree chromatic guard number χcfG(P): for all orthogonal polygon Pn with n vertices, χcfG(Pn) = O(log n) and for all polygon Pn with n vertices, χcfG(Pn) = O((log n)2).