# The independent resolving number of a graph

Mathematica Bohemica (2003)

• Volume: 128, Issue: 4, page 379-393
• ISSN: 0862-7959

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## Abstract

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For an ordered set $W=\left\{{w}_{1},{w}_{2},\cdots ,{w}_{k}\right\}$ 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 ${c}_{W}\left(v\right)=\left(d\left(v,{w}_{1}\right),d\left(v,{w}_{2}\right),\cdots ,d\left(v,{w}_{k}\right)\right).$ 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 $\mathrm{i}r\left(G\right)$. We study the existence of independent resolving sets in graphs, characterize all nontrivial connected graphs $G$ of order $n$ with $\mathrm{i}r\left(G\right)=1$, $n-1$, $n-2$, 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 $\mathrm{i}r\left(G\right)=k$ such that exactly $r$ vertices belong to every minimum independent resolving set of $G$.

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