Vertex resolvability of convex polytopes with n-paths of length p

Let be a simple, connected, and undirected graph. The distance between two vertices denoted by , is the length of the shortest path connecting u and v. A subset of vertices is said to be a resolving set for G if for any two distinct vertices V, there exist a vertex such that . A minimal resolving set is called a metric basis, and the cardinality of the basis set is called the metric dimension of G, denoted by . In this article, we find the metric dimension for two infinite families of plane graphs and , where is obtained by the combination of copies of bipartite graphs , and is obtained by the combination of double antiprism graph with antiprism graph and then adding n-paths of length p.