For an ordered set W = {w_1, w_2, ..., w_n} 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,w_1),d(v,w_2),..., d(v,w_n)). 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 SB_(m_1,m_2,...,m_n) is a graph obtained from a star graph S_n and n complete graphs K_(m_i) by merging one vertex from each K_(m_i) and the i~(th) leaf of S_n, where m_i > 3, 1 ≤ i ≤ n, and n > 2. K_m{sun}P_n graph is a graph obtained from a complete graph K_m and m copies of path graph P_n, and then joining by an edge each vertex from the i~(th) copy of P_n with the i~(th) vertex of K_m. Mobius ladder graph M_n is a graph obtained from a cycle graph C_n by connecting every pair of vertices u, v in C_n if d(u, v) = diam(C_n) for n > 5. In this paper, we determine the local metric dimension of starbarbell graph, K_m{sun}P_n graph, and Mobius ladder graph for even positive integers n > 6.
展开▼
