AbstractA set of vertices S resolves a connected graph G if every vertex is uniquelydetermined by its vector of distances to the vertices in S. The metric dimension ofG is the minimum cardinality of a resolving set of G. Let Gβ,D be the set of graphswith metric dimension β and diameter D. It is well-known that the minimum orderof a graph in Gβ,D is exactly β + D. The first contribution of this paper is tocharacterise the graphs in Gβ,D with order β + D for all values of β and D. Sucha characterisation was previously only known for D 6 2 or β 6 1. The secondcontribution is to determine the maximum order of a graph in Gβ,D for all values ofD and β. Only a weak upper bound was previously known...