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 = {s₁, s₂, …, 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.