A chromatic art gallery problem

The art gallery problem asks for the smallest number of guards required to see every point of the interior of a polygon P . We introduce and study a similar problem called the chromatic art gallery problem. Suppose that two members of a finite point guard set S ⊂ P must be given different colors if their visible regions overlap. What is the minimum number of colors required to color any guard set (not necessarily a minimal guard set) of a polygon P? We call this number, χG(P ), the chromatic guard number of P . We believe this problem has never been examined before, and it has potential applications to robotics, surveillance, sensor networks, and other areas. We show that for any spiral polygon Pspi, χG(Pspi) ≤ 2, and for any staircase polygon (strictly monotone orthogonal polygon) Psta, χG(Psta) ≤ 3. We also show that for any positive integer k, there exists a polygon Pk with 3k +2 vertices such that χG(Pk) ≥ k. Figure 1: [left] Two strictly monotone orthogonal polygons. They take the form of two vertices connected by two different staircase-shaped paths. This is the definition of “staircase polygon” that we will use. [right] Two orthogonal convex fans. This family of polygons is a subset of the strictly monotone orthogonal polygons. Some other papers use the term “staircase polygon” to refer exclusively to the orthogonal convex fans, but we will not.

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