The local metric dimension of a graph

For an ordered set W= \w₁, 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, w₁), d (v, w₂), , d (v, wₖ) ) \ 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.