The local metric dimension of starbarbell graph, $K_m\odot P_n$ graph, and M obius ladder graph

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 $r\left(v|W\right)=\left(d\right(v,w_1),d(v,w_2),\dots ,d(v,w_n\left)\right)$. 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 $S{B}_{m_1,m_2,\dots ,m_n}$ 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. $K_m\odot P_n$ 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. Möbius 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, $K_m\odot P_n$ graph, and Möbius ladder graph for even positive integers n > 6. (paper)