Optimal Multi-Level Fault-Tolerant Resolving Sets of Circulant Graph C ( n: 1, 2)

Laxman Saha, Bapan Das, Kalishankar Tiwary, Kinkar Chandra Das () and Yilun Shang ()
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Laxman Saha: Department of Mathematics, Balurghat College, Balurghat 733101, India
Bapan Das: Department of Mathematics, Balurghat College, Balurghat 733101, India
Kalishankar Tiwary: Department of Mathematics, Raiganj University, Raiganj 733134, India
Kinkar Chandra Das: Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea
Yilun Shang: Department of Computer and Information Sciences, Northumbria University, Newcastle NE1 8ST, UK

Mathematics, 2023, vol. 11, issue 8, 1-16

Abstract: Let G = ( V ( G ) , E ( G ) ) be a simple connected unweighted graph. A set R ⊂ V ( G ) is called a fault-tolerant resolving set with the tolerance level k if the cardinality of the set S x , y = { w ∈ R : d ( w , x ) ≠ d ( w , y ) } is at least k for every pair of distinct vertices x , y of G . A k -level metric dimension refers to the minimum size of a fault-tolerant resolving set with the tolerance level k . In this article, we calculate and determine the k -level metric dimension for the circulant graph C ( n : 1 , 2 ) for all possible values of k and n . The optimal fault-tolerant resolving sets with k tolerance are also delineated.

Keywords: circulant graphs; resolving set; fault-tolerant resolving set; metric dimension (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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