Different-Distance Sets in a Graph

A set of vertices S in a connected graph G is a different-distance set if, for any vertex w outside S, no two vertices in S have the same distance to w. The lower and upper different-distance number of a graph are the order of a smallest, respectively largest, maximal different-distance set. We prove that a different-distance set induces either a special type of path or an independent set. We present properties of different-distance sets, and consider the different-istance numbers of paths, cycles, Cartesian products of bipartite graphs, and Cartesian products of complete graphs. We conclude with some open problems and questions.