Abstract
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.
Highlights
The distance between two vertices u and v, denoted by dG(u, v), is the length of the shortest u − v path in a simple, undirected, connected graph G with the vertex set V(G) and the edge set E(G)
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 set S is a resolving set if distinct vertices have distinct codes, i.e., if c(x|S) = c(y|S) for all distinct x, y ∈ V(G)
Summary
The distance between two vertices u and v, denoted by dG(u, v), is the length of the shortest u − v path in a simple, undirected, connected graph G with the vertex set V(G) and the edge set E(G). If F is a fault-tolerant resolving set of C8k+r(1, 2, 3, 4) such that F ∩ I(K5i ) = ∅ and F ∩ I(K7i+−4rk+r−1) = ∅, applying Lemma 6, we have (F ∩ Ra,a+1) ∩ For every clique K5i in Cn(1, 2, 3, 4), |F| ≥ 8 − F ∩ I(K5i ) ∪ I(K7i+−4rk+r−1) , where I(Kt) denotes the set of intermediate vertices of Kt. Lemma 11. Let n ≡ 5, 6, 7, 8, 9 (mod 8) and F be a fault-tolerant resolving set of Cn(1, 2, 3, 4).
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