# Ben Phillips

## Defense

**April 11, 2002**

## Colored Distance and Competition Parameters

For a graph facility location problem, each vertex can be considered to be the location of part of a department. One seeks to optimally locate the departments in order to minimize some function of the distances between departments. Consider a plant of factory with a rectangular planar area. The area is divided into a square mesh, a grid with 1 × 1 unit pieces, and each piece is assigned to a department. Members of the same department are not required to exchange information, but they must exchange information with every other member of a different department. We seek to minimize the total distance between members of different departments. This problem is a particular instance of the colored distance problem, in which a general graph is colored with t colors and the colored distance is the sum of the distances between vertices of different colors. We show the optimal colored distance solution for the n × m grid graph when all colors are of equal size. We also consider the chromatic distance, the minimum colored distance of a proper coloring of the vertex set and examine the independent distance of a vertex, which is the maximum distance of a vertex in an independent set containing that vertex. Surprisingly, the independent set that maximizes this distance is not necessarily the one of the largest cardinality. Both the vertex independent distance of a graph decision problem and the chromatic distance of a graph decision problem are shown to be NP-complete. Furthermore, we investigate the vertex independent distance and graph independent distance for trees, cycles and grids. Finally, suppose that a vertex set S in a graph G is to be formed by two players, say the a maximizer and minimizer, alternately choosing vertices to be in S, where the resulting set must have a certain property. Here we introduce competitive processes in which one player tries to maximize the order of the resulting set S, while another player tries to minimize the order. We consider the competition parameters of competition-independence, competition-enclaveless and competition-packing. We present realizability results for the lower independence number, the minus competition-independence number and the upper independence number

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