# The independent resolving number of a graph

Gary Chartrand; Varaporn Saenpholphat; Ping Zhang

Mathematica Bohemica (2003)

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

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topChartrand, Gary, Saenpholphat, Varaporn, and Zhang, Ping. "The independent resolving number of a graph." Mathematica Bohemica 128.4 (2003): 379-393. <http://eudml.org/doc/249220>.

@article{Chartrand2003,

abstract = {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 $\mathop \{\mathrm \{i\}r\}(G)$. We study the existence of independent resolving sets in graphs, characterize all nontrivial connected graphs $G$ of order $n$ with $\mathop \{\mathrm \{i\}r\}(G) = 1$, $n-1$, $n-2$, 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 $\mathop \{\mathrm \{i\}r\}(G) = k$ such that exactly $r$ vertices belong to every minimum independent resolving set of $G$.},

author = {Chartrand, Gary, Saenpholphat, Varaporn, Zhang, Ping},

journal = {Mathematica Bohemica},

keywords = {distance; resolving set; independent set; distance; resolving set; independent set},

language = {eng},

number = {4},

pages = {379-393},

publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},

title = {The independent resolving number of a graph},

url = {http://eudml.org/doc/249220},

volume = {128},

year = {2003},

}

TY - JOUR

AU - Chartrand, Gary

AU - Saenpholphat, Varaporn

AU - Zhang, Ping

TI - The independent resolving number of a graph

JO - Mathematica Bohemica

PY - 2003

PB - Institute of Mathematics, Academy of Sciences of the Czech Republic

VL - 128

IS - 4

SP - 379

EP - 393

AB - 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 $\mathop {\mathrm {i}r}(G)$. We study the existence of independent resolving sets in graphs, characterize all nontrivial connected graphs $G$ of order $n$ with $\mathop {\mathrm {i}r}(G) = 1$, $n-1$, $n-2$, 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 $\mathop {\mathrm {i}r}(G) = k$ such that exactly $r$ vertices belong to every minimum independent resolving set of $G$.

LA - eng

KW - distance; resolving set; independent set; distance; resolving set; independent set

UR - http://eudml.org/doc/249220

ER -

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