On randomly -dimensional graphs

For an ordered set W = { w 1 , w 2 , … , w k } of vertices and a vertex v in a connected graph G , the ordered k -vector r ( v | W ) : = ( d ( v , w 1 ) , d ( v , w 2 ) , … , d ( v , w k ) ) 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 a randomly k -dimensional graph if each k -set of vertices of G is a basis of G . In this work, we study randomly k -dimensional graphs and provide some properties of these graphs.