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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:
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