The Forcing Dimension of a Graph

For an ordered set W = {w1, w2, . . . , wk} of vertices and a vertex v in a connected graph G, the (metric) representation of v with respect to W is the k-vector r(v|W ) = (d(v,w1), d(v, w2), . . . , d(v, wk)), where d(x, y) represents the distance between the vertices x and y. The setW is a resolving set for G if distinct vertices of G have distinct representations. A resolving set of minimum cardinality is a basis for G and the number of vertices in a basis is its (metric) dimension dim(G). For a basis W of G, a subset S of W is called a forcing subset of W if W is the unique basis containing S. The forcing number fG(W,dim) of W in G is the minimum cardinality of a forcing subset for W , while the forcing dimension f(G, dim) of G is the smallest forcing number among all bases of G. The forcing dimensions of some well-known graphs are determined. It is shown that for all integers a, b with 0 a b and b 1, there exists a nontrivial connected graph G with f(G) = a and dim(G) = b if and only if {a, b} = {0, 1}.