^{*}

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 repre-sentation 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 re-solving 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 ? of connected graphs is a family with constant metric dimension if dim (G) is finite and does not depend upon the choice of G in ?. In this paper, we show that dragon graph denoted by T
_{n,m} and the graph obtained from prism denoted by 2C
_{k} + {x
_{k}y
_{k}} have constant metric dimension.

If is a connected graph, the distance between two vertices is the length of a shortest path between them. Let be an ordered set of vertices of and let be a vertex of. The representation of the with respect to is denoted by is the. If distinct vertices of have distinct representations with respect to, then is called a resolving set or locating set for [

For a given ordered set of vertices of a graph, the ith component of is 0 f and only if. Thus, to show that is a resolving set it sufficient to verify that for each pair of distinct vertices .

Motivated by the problem of uniquely determining the location of an intruder in a network, the concept of metric dimension was introduced by Slater in [

By denoting the join of and, a is for and (also known as) is obtained from the by alternately deleting spokes. Caceres et al. [

Chartrand et al. proved:

Theorem 1: [

Hence paths on vertices constitute a family of graphs with constant metric dimension. Similarly, cycles with vertices also constitute such a family of graphs as their metric dimension is 2. Since are the trivalent plane graphs obtained by the cartesian product of the path with a cycle, hence they constitute a family of - with constant metric dimension. Also Javaid et al. proved in [

Let be a family of graphs of order obtained from a prism as shown in

Theorem 2: Let with , then for.

Let be a cycle with vertex set and be a path with vertex set. Dragon graph as shown in

Theorem 3: For all .

Proof of the Theorem 2: By Theorem 1,. We only need to show that is a resolving set for, which is obviously of minimal cardinality.

Case (a) When for Representations of all vertices from are as follows,

It is easy to check that all the above representations are distinct. For example, suppose that for some fixed and. Then and, a contradiction.

Case (b) When for Representations of vertices from are as follows,

All the above representations are also distinct.

Proof of the Theorem 3: By Theorem 1, . We only need to show that there is a resolving set of cardinality 2.

Case (a) When for The set is a resolving set for the graph. Representations of all vertices from are as follows,

and

,

It is easy to check that all the representations are distinct. For example, suppose that for some fixed s and j. Then because, a contradiction.

Case (b) When for The set is a resolving set for the graph. Representations of all vertices from are as follows,

and

,

All the above representations are distinct.

This research is partially supported by FAST-National University of Computer and Emerging Sciences, Peshawar, Bahauddin Zakariya University, Multan and Higher Education Commission of Pakistan.