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

Author

Listed:
  • 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)

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.

Suggested Citation

  • Laxman Saha & Bapan Das & Kalishankar Tiwary & Kinkar Chandra Das & Yilun Shang, 2023. "Optimal Multi-Level Fault-Tolerant Resolving Sets of Circulant Graph C ( n : 1, 2)," Mathematics, MDPI, vol. 11(8), pages 1-16, April.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:8:p:1896-:d:1125541
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    References listed on IDEAS

    as
    1. Laxman Saha & Mithun Basak & Kalishankar Tiwary & Kinkar Chandra Das & Yilun Shang, 2022. "On the Characterization of a Minimal Resolving Set for Power of Paths," Mathematics, MDPI, vol. 10(14), pages 1-13, July.
    2. Laxman Saha & Rupen Lama & Kalishankar Tiwary & Kinkar Chandra Das & Yilun Shang, 2022. "Fault-Tolerant Metric Dimension of Circulant Graphs," Mathematics, MDPI, vol. 10(1), pages 1-16, January.
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