Azarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21284220191201Different-Distance Sets in a Graph1511711386310.22049/cco.2019.26467.1115ENJason T. HedetniemiWingate UniversityStephen T. HedetniemiDepartment of Mathematics,
University of Johannesburg,
Auckland Park, South AfricaRenu C. Renu C. LaskarClemson UniversityHenry Martyn MulderErasmus Universiteit0000-0002-4776-4046Journal Article20180915A 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 $w$.<br />The lower and upper different-distance number of a graph are the order of a smallest, respectively largest, maximal different-distance set.<br />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-distance numbers of paths, cycles, Cartesian products of bipartite graphs, and Cartesian products of complete graphs. We conclude with some open problems and questions.http://comb-opt.azaruniv.ac.ir/article_13863_aa060ff2474ed162917d785d51209d3c.pdf