The local metric dimension of starbarbell graph, graph, and M obius ladder graph

Abstract

For an ordered set W = {w1, w2, …, wn} of n distinct vertices in a nontrivial connected graph G, the representation of a vertex v of G with respect to W is the n-vector . W is a local metric set of G if r(u|W) ≠ r(v|W) for every pair of adjacent vertices u, v in G. Local metric set with minimum cardinality is called local metric basis of G and its cardinality is the local metric dimension of G and denoted by lmd(G). Starbarbell graph is a graph obtained from a star graph Sn and n complete graphs by merging one vertex from each and the ith leaf of Sn, where mi > 3, 1 ≤ i ≤ n, and n > 2. graph is a graph obtained from a complete graph Km and m copies of path graph Pn, and then joining by an edge each vertex from the ith copy of Pn with the ith vertex of Km. Möbius ladder graph Mn is a graph obtained from a cycle graph Cn by connecting every pair of vertices u, v in Cn if d(u, v) = diam(Cn) for n > 5. In this paper, we determine the local metric dimension of starbarbell graph, graph, and Möbius ladder graph for even positive integers n > 6.