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Title: The directed distance dimension of oriented graphs (English)
Author: Chartrand, Gary
Author: Raines, Michael
Author: Zhang, Ping
Language: English
Journal: Mathematica Bohemica
ISSN: 0862-7959
Volume: 125
Issue: 2
Year: 2000
Pages: 155-168
Summary lang: English
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Category: math
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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 $r(v\bigm|W) = ( d(v, w_1), d(v, w_2), \cdots, d(v, w_k) )$, 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 $\dim(D) \leq n-3$ 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
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Date available: 2009-09-24T21:41:50Z
Last updated: 2015-09-15
Stable URL: http://hdl.handle.net/10338.dmlcz/125961
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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), 191-195. MR 0457289
Reference: [3] P. J. Slater: Leaves of trees.Congress. Numer. 11, (1975), 549-559. Zbl 0316.05102, MR 0422062
Reference: [4] P. J. Slater: Dominating and reference sets in graphs.J. Math. Phys. Sci. 22 (1988), 445-455. MR 0966610
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