Fault-Tolerant Metric Dimension of Circulant Graphs

Let G be a connected graph with vertex set V(G) and d(u,v) be the distance between the vertices u and v. A set of vertices S={s(1),s(2), horizontal ellipsis ,s(k)}& SUB;V(G) is called a resolving set for G if, for any two distinct vertices u,v & ISIN;V(G), there is a vertex s(i)& ISIN;S such that d(u,s(i))& NOTEQUAL;d(v,s(i)). A resolving set S for G is fault-tolerant if S\{x} is also a resolving set, for each x in S, and the fault-tolerant metric dimension of G, denoted by beta & PRIME;(G), is the minimum cardinality of such a set. The paper of Basak et al. on fault-tolerant metric dimension of circulant graphs C-n(1,2,3) has determined the exact value of beta & PRIME;(C-n(1,2,3)). In this article, we extend the results of Basak et al. to the graph C-n(1,2,3,4) and obtain the exact value of beta & PRIME;(C-n(1,2,3,4)) for all n & GE;22.