On lower bounds for the metric dimension of graphs

For an ordered set W = {w1, w2, . . . , wk} of vertices and a vertex v in a connected graph G, the ordered k-vector r(v|W ) = (d(v, w1), d(v, w2), . . . , d(v, wk)) is called the (metric) representation of v with respect to W , where d(x, y) is the distance between the vertices x and y. A set W is called a resolving set for G if distinct vertices of G have distinct representations with respect to W . The minimum car-dinality of a resolving set for G is its metric dimension, and a resolving set of minimum cardinality is a basis of G. Lower bounds for metric di-mension are important. In this paper, we investigate lower bounds for metric dimension. Motivated by a lower bound for the metric dimension k of a graph of order n with diameter d in [S. Khuller, B. Raghavachari, and A. Rosenfeld, Landmarks in graphs, Discrete Applied Mathematics 70(3)(1996)217 − 229], which states that k ≥ n − dk, we characterize all graphs with this lower bound and obtain a new lower bound. This new bound is better than the previous one, for graphs with diameter more than 3.