On randomly k-dimensional 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. The set W is called a resolving set for G if distinct vertices of G have distinct representations with respect to W . A resolving set for G with minimum cardinality is called a basis of G and its cardinality is the metric dimension of G. A connected graph G is called randomly k-dimensional graph if each k-set of vertices of G is a basis of G. In this paper, we study randomly k-dimensional graphs and provide some properties of these graphs.