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Fault-Tolerant Resolvability and Extremal Structures of Graphs

Author

Listed:
  • Hassan Raza

    (School of Mathematical Sciences, Anhui University, Hefei 230601, China)

  • Sakander Hayat

    (Faculty of Engineering Sciences, GIK Institute of Engineering Sciences and Technology, Topi, Swabi 23460, Pakistan)

  • Muhammad Imran

    (Department of Mathematical Sciences, United Arab Emirates University, Al Ain 15551, UAE
    School of Natural Sciences, National University of Sciences and Technology, H-12, Islamabad 44000, Pakistan)

  • Xiang-Feng Pan

    (School of Mathematical Sciences, Anhui University, Hefei 230601, China)

Abstract

In this paper, we consider fault-tolerant resolving sets in graphs. We characterize n -vertex graphs with fault-tolerant metric dimension n , n − 1 , and 2, which are the lower and upper extremal cases. Furthermore, in the first part of the paper, a method is presented to locate fault-tolerant resolving sets by using classical resolving sets in graphs. The second part of the paper applies the proposed method to three infinite families of regular graphs and locates certain fault-tolerant resolving sets. By accumulating the obtained results with some known results in the literature, we present certain lower and upper bounds on the fault-tolerant metric dimension of these families of graphs. As a byproduct, it is shown that these families of graphs preserve a constant fault-tolerant resolvability structure.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:1:p:78-:d:197365
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    References listed on IDEAS

    as
    1. Yilun Shang, 2018. "Resilient Multiscale Coordination Control against Adversarial Nodes," Energies, MDPI, vol. 11(7), pages 1-17, July.
    2. Raza, Hassan & Hayat, Sakander & Pan, Xiang-Feng, 2018. "On the fault-tolerant metric dimension of convex polytopes," Applied Mathematics and Computation, Elsevier, vol. 339(C), pages 172-185.
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    Cited by:

    1. 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).
    2. Juan Wang & Lianying Miao & Yunlong Liu, 2019. "Characterization of n -Vertex Graphs of Metric Dimension n − 3 by Metric Matrix," Mathematics, MDPI, vol. 7(5), pages 1-13, May.
    3. Sakander Hayat & Asad Khan & Yubin Zhong, 2022. "On Resolvability- and Domination-Related Parameters of Complete Multipartite Graphs," Mathematics, MDPI, vol. 10(11), pages 1-16, May.

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