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Hedetniemi, Jason T., Hedetniemi, Stephen T., Renu C. Laskar, Renu C., Mulder, Henry Martyn. (1398). Different-Distance Sets in a Graph. , 4(2), 151-171. doi: 10.22049/cco.2019.26467.1115
Jason T. Hedetniemi; Stephen T. Hedetniemi; Renu C. Renu C. Laskar; Henry Martyn Mulder. "Different-Distance Sets in a Graph". , 4, 2, 1398, 151-171. doi: 10.22049/cco.2019.26467.1115
Hedetniemi, Jason T., Hedetniemi, Stephen T., Renu C. Laskar, Renu C., Mulder, Henry Martyn. (1398). 'Different-Distance Sets in a Graph', , 4(2), pp. 151-171. doi: 10.22049/cco.2019.26467.1115
Hedetniemi, Jason T., Hedetniemi, Stephen T., Renu C. Laskar, Renu C., Mulder, Henry Martyn. Different-Distance Sets in a Graph. , 1398; 4(2): 151-171. doi: 10.22049/cco.2019.26467.1115

Different-Distance Sets in a Graph

Communications in Combinatorics and Optimization
مقاله 8، دوره 4، شماره 2، اسفند 2019، صفحه 151-171    اصل مقاله (586.58 K)
نوع مقاله: Original paper
شناسه دیجیتال (DOI): 10.22049/cco.2019.26467.1115
نویسندگان
Jason T. Hedetniemi* 1؛ Stephen T. Hedetniemi2؛ Renu C. Renu C. Laskar3؛ Henry Martyn Mulder4
1Wingate University
2Department of Mathematics, University of Johannesburg, Auckland Park, South Africa
3Clemson University
4Erasmus Universiteit
چکیده
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.
کلیدواژه‌ها
Di fferent-Distance Sets؛ Cartesian products؛ graph
مراجع
[1] G. Chartrand, L. Eroh, M.A. Johnson, and O.R. Oellermann, Resolvability in graphs and the metric dimension of a graph, Discrete Appl. Math. 105 (2000), no. 1-3, 99–113.

[2] G. Chartrand and P. Zhang, The theory and application of resolvability in graphs, Congr. Numer. 160 (2003), 47–68.

[3] R. Hammack, W. Imrich, and S. Klavzar, Handbook of graph products, CRC Press Boca Raton FL, 2011.

[4] H.M. Mulder, The interval function of a graph, Math. Centre Tracts 132, Amsterdam, 1980.

[5] P.J. Slater, Leaves of trees, Congr. Numer. 14 (1975), 549–559.

 

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