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.