The interval function (in the sense of H. M. Mulder) is an important tool for studying those properties of a connected graph that depend on the distance between vertices. An axiomatic characterization of the interval function of a connected graph was published by Nebeský in 1994. In Section 2 of the present paper, a simpler and shorter proof of that characterization will be given. In Section 3, a characterization of geodetic graphs will be established; this characterization will utilize properties of the interval function.
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 6 i 6 k. The set W is a local metric set of G if code(u) 6= 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.