On randomly kk-dimensional graphs

For an ordered set W={w1,w2,…,wk}W={w1,w2,…,wk} of vertices and a vertex vv in a connected graph GG, the ordered kk-vector r(v|W):=(d(v,w1),d(v,w2),…,d(v,wk))r(v|W):=(d(v,w1),d(v,w2),…,d(v,wk)) is called the (metric) representation of vv with respect to WW, where d(x,y)d(x,y) is the distance between the vertices xx and yy. The set WW is called a resolving set for GG if distinct vertices of GG have distinct representations with respect to WW. A resolving set for GG with minimum cardinality is called a basis of GG and its cardinality is the metric dimension of GG. A connected graph GG is called a randomly kk-dimensional graph if each kk-set of vertices of GG is a basis of GG. In this work, we study randomly kk-dimensional graphs and provide some properties of these graphs.