On the Forcing Dimension of a Graph

A set W subset of V(G) is called a resolving set, if for each two distinct vertices u,v is an element of V(G) there exists w is an element of W such that d(u,w) not equal d(v, w), where d(x, y) is the distance between the vertices x and y. A resolving set for G with minimum cardinality is called a metric basis. The forcing dimension f (G, dim) (or f (G)) of G is the smallest cardinality of a subset S subset of V (G) such that there is a unique basis containing S. The forcing dimensions of some well-known graphs are determined. In this paper, among some other results, it is shown that for large enough integer n and all integers a, b with 0 <= a <= b < n and b >= 1, there exists a nontrivial connected graph G of order n with f(C) = a and dim(G) = b if {a, b} not equal {0,1}.