Different-distance sets in a graph

• Main Authors: Jason T. Hedetniemi, Stephen T. Hedetniemi, Renu C. Laskar, Henry Martyn Mulder4
• Format: Article
• Language: English
• Published: Azarbaijan Shahide Madani University 2019-12-01
• Series: Communications in Combinatorics and Optimization
• Subjects: different-distance set; different-distance number
• Online Access: http://comb-opt.azaruniv.ac.ir/article_13863_aa060ff2474ed162917d785d51209d3c.pdf

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-distance numbers of paths, cycles, Cartesian products of bipartite graphs, and Cartesian products of complete graphs. We conclude with some open problems and questions.