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 , … , s k } ⊂ V ( G ) is called a resolving set for G if, for any two distinct vertices u , v ∈ V ( G ) , there is a vertex s i ∈ S such that d ( u , s i ) ≠ 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 β ′ ( 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 β ′ ( 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 β ′ ( C n ( 1 , 2 , 3 , 4 ) ) for all n ≥ 22 .