TY - JOUR

T1 - Fault-Tolerant Metric Dimension of Circulant Graphs

AU - Saha, Laxman

AU - Lama, Rupen

AU - Tiwary, Kalishankar

AU - Das, Kinkar Chandra

AU - Shang, Yilun

N1 - Funding information: L. Saha is supported by Science and Research Board (SERB), DST, India (Grant No. CRG/2019/006909). K. C. Das is supported by National Research Foundation funded by the Korean government (Grant No. 2021R1F1A1050646).

PY - 2022/1/1

Y1 - 2022/1/1

N2 - 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.

AB - 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.

KW - circulant graphs

KW - resolving set

KW - fault-tolerant resolving set

KW - fault-tolerant metric dimension

U2 - 10.3390/math10010124

DO - 10.3390/math10010124

M3 - Article

SN - 2227-7390

VL - 10

JO - Mathematics

JF - Mathematics

IS - 1

M1 - e124

ER -