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Characterization of n -Vertex Graphs of Metric Dimension n − 3 by Metric Matrix

Author

Listed:
  • Juan Wang

    (School of Mathematics, China University of Mining and Technology, Xuzhou 221116, China)

  • Lianying Miao

    (School of Mathematics, China University of Mining and Technology, Xuzhou 221116, China)

  • Yunlong Liu

    (College of Information and Control Engineering, Weifang University, Weifang 261061, China)

Abstract

Let G = ( V ( G ) , E ( G ) ) be a connected graph. An ordered set W ⊂ V ( G ) is a resolving set for G if every vertex of G is uniquely determined by its vector of distances to the vertices in W . The metric dimension of G is the minimum cardinality of a resolving set. In this paper, we characterize the graphs of metric dimension n − 3 by constructing a special distance matrix, called metric matrix. The metric matrix makes it so a class of graph and its twin graph are bijective and the class of graph is obtained from its twin graph, so it provides a basis for the extension of graphs with respect to metric dimension. Further, the metric matrix gives a new idea of the characterization of extremal graphs based on metric dimension.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:5:p:479-:d:234532
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    References listed on IDEAS

    as
    1. András Sebő & Eric Tannier, 2004. "On Metric Generators of Graphs," Mathematics of Operations Research, INFORMS, vol. 29(2), pages 383-393, May.
    2. 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.
    3. Kelenc, Aleksander & Kuziak, Dorota & Taranenko, Andrej & G. Yero, Ismael, 2017. "Mixed metric dimension of graphs," Applied Mathematics and Computation, Elsevier, vol. 314(C), pages 429-438.
    4. Jia-Bao Liu & Agha Kashif & Tabasam Rashid & Muhammad Javaid, 2019. "Fractional Metric Dimension of Generalized Jahangir Graph," Mathematics, MDPI, vol. 7(1), pages 1-10, January.
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    Cited by:

    1. Wahyuni Abidin & Anm Salman & Suhadi Wido Saputro, 2022. "Non-Isolated Resolving Sets of Corona Graphs with Some Regular Graphs," Mathematics, MDPI, vol. 10(6), pages 1-13, March.

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