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Optimal Fault-Tolerant Resolving Set of Power Paths

Author

Listed:
  • Laxman Saha

    (Department of Mathematics, Balurghat College, Balurghat 733101, India)

  • Rupen Lama

    (Department of Mathematics, Balurghat College, Balurghat 733101, India)

  • Bapan Das

    (Department of Mathematics, Balurghat College, Balurghat 733101, India)

  • Avishek Adhikari

    (Department of Mathematics, Presidency University, Kolkata 700073, India)

  • Kinkar Chandra Das

    (Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea)

Abstract

In a simple connected undirected graph G , an ordered set R of vertices is called a resolving set if for every pair of distinct vertices u and v , there is a vertex w ∈ R such that d ( u , w ) ≠ d ( v , w ) . A resolving set F for the graph G is a fault-tolerant resolving set if for each v ∈ F , F ∖ { v } is also a resolving set for G . In this article, we determine an optimal fault-resolving set of r -th power of any path P n when n ≥ r ( r − 1 ) + 2 . For the other values of n , we give bounds for the size of an optimal fault-resolving set. We have also presented an algorithm to construct a fault-tolerant resolving set of P m r from a fault-tolerant resolving set of P n r where m < n .

Suggested Citation

  • Laxman Saha & Rupen Lama & Bapan Das & Avishek Adhikari & Kinkar Chandra Das, 2023. "Optimal Fault-Tolerant Resolving Set of Power Paths," Mathematics, MDPI, vol. 11(13), pages 1-18, June.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:13:p:2868-:d:1179941
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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.
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