On two families of 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 \(W = {w_1, w_2, …., w_k}\) 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.