**Journal of Prime Research in Mathematics**

Vol. 1 (2012), Issue 1, pp. 92 – 101

ISSN: 1817-3462 (Online) 1818-5495 (Print)

ISSN: 1817-3462 (Online) 1818-5495 (Print)

# On two families of graphs with constant metric dimension

**M. Ali
**Department of Mathematics, National University of Computer & Emerging Sciences, FAST, Peshawar, Pakistan.

**Department of Mathematics, National University of Computer & Emerging Sciences, FAST, Peshawar, Pakistan.**

M. T. Rahim

M. T. Rahim

**G. Ali**

Department of Mathematics, National University of Computer & Emerging Sciences, FAST, Peshawar, Pakistan.

\(^{1}\)Corresponding Author: murtaza psh@yahoo.com

Copyright © 2012 M. Ali, M. T. Rahim, G. Ali. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Published:**December, 2012.

### Abstract

If \(G\) is a connected graph, the distance d(u, v) between two vertices \(u, v ∈ V (G)\) is the length of a shortest path between them. Let \(W = {w_1, w_2, …., w_k}\) be an ordered set of vertices of \(G\) and let \(v\) be a vertex of \(G\). The representation r(v|W) of \(v\) with respect to \(W\) is the k-tuple \((d(v, w_1), d(v, w_2), ….., d(v, w_k))\). If distinct vertices of \(G\) have distinct representations with respect to W, then W is called a resolving set or locating set for \(G\). A resolving set of minimum cardinality is called a basis for \(G\) and this cardinality is the metric dimension of \(G\), denoted by \(dim(G)\). A family G of connected graphs is a family with constant metric dimension if \(dim(G)\) does not depend upon the choice of \(G\) in \(G\). In this paper, we show that the graphs (D^{∗}_{p}\) and \(D^{n}_{p}\), obtained from prism graph have constant metric dimension.

#### Keywords:

Metric dimension, basis, resolving set, prism.