The theory of location of facilities in networks combines tools from graph theory, basic analysis, optimization, and complexity theory. The central issue is the study of the optimal location(s) of a facility such as emergency installation, a supply depot, a switching center, a pumping station, an obnoxious dump, a communications center, or the like in a network such as a street or road network, an electrical network, a network of channels or pipes, a communications network or the like. Optimality depends on criteria usually involving some idea of distance and varies according to the application. Weighted graphs, often referred to as networks, provide a context for studying these types of problems, where vertices and edges are assigned weights representing certain parameters according to the application. Usually, special sets of points in the network are sought that are either "central" or "peripheral." Results range from the descriptions of optimal locations to the computational difficulty in actually determining these optimal locations. Considerable study has been focused on weighted trees. These issues have motivated graph theorists to probe many different notions of centrality and notions of the "outer fringes" in ordinary (unweighted) graphs, particularly trees. In such models, users and facility locations are thought to be restricted to vertices. However, the graph theoretical origins of centrality precede the advent of modern location theory as C. Jordan introduced the concepts of the center of a tree and the branch weight centroid of a tree in 1869.
We will survey results concerning the structure of several "central sets" of vertices in trees, beginning with the center, the branch-weight centroid, and the median. We will discuss the (distance) balanced vertices, several one-parameter families of central sets, a two-parameter family of central sets, the cutting center, and the security centroid.

Thursday, 31 January 2002
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)