On the strong metric dimension of Cartesian and direct products of graphs
Abstract
Let $G$ be a connected graph. A vertex $w$ {\em strongly resolves} a pair $u, v$ of vertices of $G$ if there exists some shortest $u-w$ path containing $v$ or some shortest $v-w$ path containing $u$. A set $W$ of vertices is a {\em strong resolving set} for $G$ if every pair of vertices of $G$ is strongly resolved by some vertex of $W$. The smallest cardinality of a strong resolving set for $G$ is called the {\em strong metric dimension} of $G$. It is known that the problem of computing the strong metric dimension of a graph is NP-hard. In this paper we obtain closed formulae for the strong metric dimension of several families of Cartesian product graphs and direct product graphs.
- Publication:
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arXiv e-prints
- Pub Date:
- July 2013
- DOI:
- 10.48550/arXiv.1307.4722
- arXiv:
- arXiv:1307.4722
- Bibcode:
- 2013arXiv1307.4722R
- Keywords:
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- Mathematics - Combinatorics;
- 05C12;
- 05C76;
- 05C90
- E-Print:
- 17 pages