Journal of Prime Research in Mathematics

# On two families of graphs with constant metric dimension

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

$$^{1}$$Corresponding Author: murtaza psh@yahoo.com

### 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.