A CHARACTERIZATION FOR METRIC TWO-DIMENSIONAL GRAPHS AND THEIR ENUMERATION

The metric dimension of a connected graph G is the minimum number of vertices in a subset B of G such that all other vertices are uniquely determined by their distances to the vertices in B. In this case, B is called a metric basis for G. The basic distance of a metric two-dimensional graph G is the distance between the elements of B. Giving a characterization for those graphs whose metric dimensions are two, we enumerate the number of n vertex metric two-dimensional graphs with the basic distance 1.