Fault-Tolerant Metric Dimension of Circulant Graphs

Let G be a connected graph with vertex set V(G) and d(u,v) the distance between vertices u v. A of S={s1,s2,…,sk}?V(G) is called resolving for if, any two distinct u,v?V(G), there si?S such that d(u,si)?d(v,si). S fault-tolerant if S\{x} also set, each x in S, metric dimension G, denoted by ??(G), minimum cardinality set. The paper Basak et al. on circulant graphs Cn(1,2,3) has determined exact value ??(Cn(1,2,3)). In this article, we extend results to Cn(1,2,3,4) obtain ??(Cn(1,2,3,4)) all n?22.