The local metric dimension of a graph

For an ordered set W= ₁,w₂,… ,wₖ\ of k distinct vertices in a nontrivial connected graph G, the metric code of a vertex v of G with respect to W is the k-vector \[ { code}(v)= ( d(v,w_1),d(v,w_2),⋯ ,d(v,w_k) ) \] where d(v,wᵢ) is the distance between v and wᵢ for 1≤ i≤ k. The set W is a local metric set of G if code(u)≠ code(v) for every pair $u,v$ of adjacent vertices of G. The minimum positive integer k for which G has a local metric k-set is the local metric dimension lmd(G) of G. A local metric set of G of cardinality lmd(G) is a local metric basis of G. We characterize all nontrivial connected graphs of order n having local metric dimension $1$, $n-2$, or $n-1$ and establish sharp bounds for the local metric dimension of a graph in terms of well-known graphical parameters. Several realization results are presented along with other results on the number of local metric bases of a connected graph.