Title:

The independent resolving number of a graph (English) 
Author:

Chartrand, G. 
Author:

Saenpholphat, V. 
Author:

Zhang, P. 
Language:

English 
Journal:

Mathematica Bohemica 
ISSN:

08627959 (print) 
ISSN:

24647136 (online) 
Volume:

128 
Issue:

4 
Year:

2003 
Pages:

379393 
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 $\mathop {\mathrm ir}(G) = 1$, $n1$, $n2$, 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:

20090924T22:10:59Z 
Last updated:

20200729 
Stable URL:

http://hdl.handle.net/10338.dmlcz/134003 
. 
Reference:

[1] F. Buckley, F. Harary: Distance in Graphs.AddisonWesley, 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 
. 