The Dimensions of Graphs and Constructions of Their Bases

Let G= (V,E) be a connected graph and W = {w1 ,w1,… ,wk } an ordered set of V. Given v ∈ V, the repsolving set of G if r(u|W)=r(v|W) implies that u=v for all pairs {u,v} of vertices of G. The resolving set of G with the smallest cardinality is called a basis of G. The dimension of G, dim (G), is the cardinality of a basis for G. The bound of a Cartesian product of a connected graph H and a path Pk was reached: dim(H)≤dim (H × Pk)≤dim (H)+1. Then, the dimension value of some graphs was given. At last, the constructions of some graphs'bases were showed.