Resolvability in circulant graphs

A set W of the vertices of a connected graph G is called a resolving set for G if for every two distinct vertices u, . . V (G) there is a vertex w . W such that d(u,w) . d(.,w). A resolving set of minimum cardinality is called a metric basis for G and the number of vertices in a metric basis is called the metric dimension of G, denoted by dim(G). For a vertex u of G and a subset S of V (G), the distance between u and S is the number mins.S d(u, s). A k-partition . = {S 1, S 2, ., S k } of V (G) is called a resolving partition if for every two distinct vertices u, v . V (G) there is a set S i in . such that d(u, S i ) . d(v, S i ). The minimum k for which there is a resolving k-partition of V (G) is called the partition dimension of G, denoted by pd(G). The circulant graph is a graph with vertex set . n , an additive group of integers modulo n, and two vertices labeled i and j adjacent if and only if i . j (mod n) . C, where C . . n has the property that C = .C and 0 . C. The circulant graph is denoted by X n,. where . = |C|. In this paper, we study the metric dimension of a family of circulant graphs X n,3 with connection set C={1,n2,n.1} and prove that dim(X n,3) is independent of choice of n by showing that dim(Xn,3)={3foralln.0(mod4),4foralln.2(mod4). We also study the partition dimension of a family of circulant graphs X n,4 with connection set C = {±1,±2} and prove that pd(X n,4) is independent of choice of n and show that pd(X 5,4) = 5 and pd(Xn,4)={3foralloddn.9,4forallevenn.6andn=7.pd(Xn,4)={3foralloddn. 9,4forallevenn.6andn=7.