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Twin vertices in fault-tolerant metric sets and fault-tolerant metric dimension of multistage interconnection networks

Author

Listed:
  • Prabhu, S.
  • Manimozhi, V.
  • Arulperumjothi, M.
  • Klavžar, Sandi

Abstract

A set of vertices S⊆V(G) is a resolving set of a graph G if for each x,y∈V(G) there is a vertex u∈S such that d(x,u)≠d(y,u). A resolving set S is a fault-tolerant resolving set if S∖{x} is a resolving set for every x∈S. The fault-tolerant metric dimension (FTMD) β′(G) of G is the minimum cardinality of a fault-tolerant resolving set. It is shown that each twin vertex of G belongs to every fault-tolerant resolving set of G. As a consequence, β′(G)=n(G) iff each vertex of G is a twin vertex, which corrects a wrong characterization of graphs G with β′(G)=n(G) from [Mathematics 7(1) (2019) 78]. This FTMD problem is reinvestigated for Butterfly networks, Benes networks, and silicate networks. This extends partial results from [IEEE Access 8 (2020) 145435–145445], and at the same time, disproves related conjectures from the same paper.

Suggested Citation

  • Prabhu, S. & Manimozhi, V. & Arulperumjothi, M. & Klavžar, Sandi, 2022. "Twin vertices in fault-tolerant metric sets and fault-tolerant metric dimension of multistage interconnection networks," Applied Mathematics and Computation, Elsevier, vol. 420(C).
    • Handle: RePEc:eee:apmaco:v:420:y:2022:i:c:s0096300321009802
    DOI: 10.1016/j.amc.2021.126897
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    References listed on IDEAS

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    1. Hassan Raza & Sakander Hayat & Muhammad Imran & Xiang-Feng Pan, 2019. "Fault-Tolerant Resolvability and Extremal Structures of Graphs," Mathematics, MDPI, vol. 7(1), pages 1-19, January.
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    Cited by:

    1. Arulperumjothi, M. & Klavžar, Sandi & Prabhu, S., 2023. "Redefining fractal cubic networks and determining their metric dimension and fault-tolerant metric dimension," Applied Mathematics and Computation, Elsevier, vol. 452(C).

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