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={s1,s2,…,sk}⊂V(G) is called a resolving set for G if, for any two distinct vertices u,v∈V(G), there is a vertex si∈S such that d(u,si)≠d(v,si). 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 β′(G), is the minimum cardinality of such a set. The paper of Basak et al. on fault-tolerant metric dimension of circulant graphs Cn(1,2,3) has determined the exact value of β′(Cn(1,2,3)). In this article, we extend the results of Basak et al. to the graph Cn(1,2,3,4) and obtain the exact value of β′(Cn(1,2,3,4)) for all n≥22.

In a simple connected undirected graph G, an ordered set R of vertices is called a resolving set if for every pair of distinct vertices u and v, there is a vertex w∈R such that d(u,w)≠d(v,w). A resolving set F for the graph G is a fault-tolerant resolving set if for each v∈F, F∖{v} is also a resolving set for G. In this article, we determine an optimal fault-resolving set of r-th power of any path Pn when n≥r(r−1)+2. For the other values of n, we give bounds for the size of an optimal fault-resolving set. We have also presented an algorithm to construct a fault-tolerant resolving set of Pmr from a fault-tolerant resolving set of Pnr where m<n.