On Cycle Related Graphs with Constant Metric Dimension

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 = {w1, w2, ..., wk} 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,w1), d(v,w2), …, d(v,wk)). . 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 Tn,m and the graph obtained from prism denoted by 2Ck + {xkyk} have constant metric dimension.