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 r(v|W) = (d(v, w1), d(v, w2), ..., d(v, wn)). 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 SBm1,m2,...,mn is a graph obtained from a star graph Sn and n complete graphs Kmi by merging one vertex from each Kmi and the i th leaf of Sn, where mi ≥ 3, 1 ≤ i ≤ n, and n ≥ 2. Km ? Pn 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 i th copy of Pn with the i th vertex of Km. M¨obius 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, Km ? Pn graph, and M¨obius ladder graph for even positive integers n ≥ 6.