Title:

The directed distance dimension of oriented graphs (English) 
Author:

Chartrand, Gary 
Author:

Raines, Michael 
Author:

Zhang, Ping 
Language:

English 
Journal:

Mathematica Bohemica 
ISSN:

08627959 
Volume:

125 
Issue:

2 
Year:

2000 
Pages:

155168 
Summary lang:

English 
. 
Category:

math 
. 
Summary:

For a vertex $v$ of a connected oriented graph $D$ and an ordered set $W = \{w_1,w_2,\cdots,w_k\}$ of vertices of $D$, the (directed distance) representation of $v$ with respect to $W$ is the ordered $k$tuple $r(v\bigmW) = ( d(v, w_1), d(v, w_2), \cdots, d(v, w_k) )$, where $d(v, w_i)$ is the directed distance from $v$ to $w_i$. The set $W$ is a resolving set for $D$ if every two distinct vertices of $D$ have distinct representations. The minimum cardinality of a resolving set for $D$ is the (directed distance) dimension $\dim(D)$ of $D$. The dimension of a connected oriented graph need not be defined. Those oriented graphs with dimension 1 are characterized. We discuss the problem of determining the largest dimension of an oriented graph with a fixed order. It is shown that if the outdegree of every vertex of a connected oriented graph $D$ of order $n$ is at least 2 and $\dim(D)$ is defined, then $\dim(D) \leq n3$ and this bound is sharp. (English) 
Keyword:

oriented graphs 
Keyword:

directed distance 
Keyword:

resolving sets 
Keyword:

dimension 
MSC:

05C12 
MSC:

05C20 
idZBL:

Zbl 0963.05045 
idMR:

MR1768804 
. 
Date available:

20090924T21:41:50Z 
Last updated:

20150915 
Stable URL:

http://hdl.handle.net/10338.dmlcz/125961 
. 
Reference:

[1] G. Chartrand L. Eroh M. Johnson O. R. Oellermann: Resolvability in graphs and the metric dimension of a graph.Preprint. MR 1780464 
Reference:

[2] F. Harary R. A. Melter: On the metric dimension of a graph.Ars Combin. 2 (1976), 191195. MR 0457289 
Reference:

[3] P. J. Slater: Leaves of trees.Congress. Numer. 11, (1975), 549559. Zbl 0316.05102, MR 0422062 
Reference:

[4] P. J. Slater: Dominating and reference sets in graphs.J. Math. Phys. Sci. 22 (1988), 445455. MR 0966610 
. 