Fault-Tolerant Metric Dimension of Circulant Graphs - Laxman Saha; Rupen Lama; Kalishankar Tiwary; Kinkar Chandra Das; Yilun Shang (Mathematics, 2022)
Let G be a connected graph with vertex set <semantics>V(G)</semantics> and <semantics>d(u,v)</semantics> be the distance between the vertices u and v. A set of vertices <semantics>S={s1,s2,…,sk}⊂V(G)</semantics> is called a resolving set for G if, for any two distinct vertices <semantics>u,v∈V(G)</semantics>, there is a vertex <semantics>si∈S</semantics> such that <semantics>d(u,si)≠d(v,si)</semantics>. A resolving set S for G is fault-tolerant if <semantics>S\{x}</semantics> is also a resolving set, for each x in S, and the fault-tolerant metric dimension of G, denoted by <semantics>β′(G)</semantics>, is the minimum cardinality of such a set. The paper of Basak et al. on fault-tolerant metric dimension of circulant graphs <semantics>Cn(1,2,3)</semantics> has determined the exact value of <semantics>β′(Cn(1,2,3))</semantics>. In this article, we extend the results of Basak et al. to the graph <semantics>Cn(1,2,3,4)</semantics> and obtain the exact value of <semantics>β′(Cn(1,2,3,4))</semantics> for all <semantics>n≥22</semantics>.