Distribution Centers in Graphs

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For a graph G=(V,E) and a set S⊆V, the boundary of S is the set of vertices in V∖S that have a neighbor in S. A non-empty set S⊆V is a distribution center if for every vertex v in the boundary of S, v is adjacent to a vertex in S, say u, where u has at least as many neighbors in S as v has in V∖S. The distribution center number of a graph G is the minimum cardinality of a distribution center of G. We introduce distribution centers as graph models for supply–demand type distribution. We determine the distribution center number for selected families of graphs and give bounds on the distribution center number for general graphs. Although not necessarily true for general graphs, we show that for trees the domination number and the maximum degree are upper bounds on the distribution center number.