# Article

 Title: The independent resolving number of a graph (English) Author: Chartrand, G. Author: Saenpholphat, V. Author: Zhang, P. Language: English Journal: Mathematica Bohemica ISSN: 0862-7959 (print) ISSN: 2464-7136 (online) Volume: 128 Issue: 4 Year: 2003 Pages: 379-393 Summary lang: English . Category: math . Summary: For an ordered set $W=\lbrace w_1, w_2, \dots , w_k\rbrace$ of vertices in a connected graph $G$ and a vertex $v$ of $G$, the code of $v$ with respect to $W$ is the $k$-vector $c_W(v) = (d(v, w_1), d(v, w_2), \dots , d(v, w_k) ).$ The set $W$ is an independent resolving set for $G$ if (1) $W$ is independent in $G$ and (2) distinct vertices have distinct codes with respect to $W$. The cardinality of a minimum independent resolving set in $G$ is the independent resolving number $\mathop {\mathrm ir}(G)$. We study the existence of independent resolving sets in graphs, characterize all nontrivial connected graphs $G$ of order $n$ with $\mathop {\mathrm ir}(G) = 1$, $n-1$, $n-2$, and present several realization results. It is shown that for every pair $r, k$ of integers with $k \ge 2$ and $0 \le r \le k$, there exists a connected graph $G$ with $\mathop {\mathrm ir}(G) = k$ such that exactly $r$ vertices belong to every minimum independent resolving set of $G$. (English) Keyword: distance Keyword: resolving set Keyword: independent set MSC: 05C12 MSC: 05C69 idZBL: Zbl 1050.05043 idMR: MR2032475 DOI: 10.21136/MB.2003.134003 . Date available: 2009-09-24T22:10:59Z Last updated: 2020-07-29 Stable URL: http://hdl.handle.net/10338.dmlcz/134003 . Reference: [1] F. Buckley, F. Harary: Distance in Graphs.Addison-Wesley, Redwood City, CA, 1990. MR 1045632 Reference: [2] G. Chartrand, L. Lesniak: Graphs $\&$ Digraphs, third edition.CRC Press, Boca Raton, 1996. MR 1408678 Reference: [3] F. Harary, R. A. Melter: On the metric dimension of a graph.Ars Combin. 2 (1976), 191–195. MR 0457289 Reference: [4] P. J. Slater: Leaves of trees.Congress. Numer. 14 (1975), 549–559. Zbl 0316.05102, MR 0422062 Reference: [5] P. J. Slater: Dominating and reference sets in graphs.J. Math. Phys. Sci. 22 (1988), 445–455. MR 0966610 .

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