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Metric Dimensions of Bicyclic Graphs

Author

Listed:
  • Asad Khan

    (School of Computer Science and Cyber Engineering, Guangzhou University, Guangzhou 510006, China
    These authors contributed equally to this work.)

  • Ghulam Haidar

    (Department of Mathematics and Statistics, The University of Haripur, Haripur 22620, Pakistan
    These authors contributed equally to this work.)

  • Naeem Abbas

    (Department of Mathematics and Statistics, The University of Haripur, Haripur 22620, Pakistan
    These authors contributed equally to this work.)

  • Murad Ul Islam Khan

    (Department of Mathematics and Statistics, The University of Haripur, Haripur 22620, Pakistan
    These authors contributed equally to this work.)

  • Azmat Ullah Khan Niazi

    (Department of Mathematics and Statistics, The University of Lahore, Sargodha 40100, Pakistan)

  • Asad Ul Islam Khan

    (Economics Department, Ibn Haldun University, Istanbul 34480, Turkey)

Abstract

The distance d ( v a , v b ) between two vertices of a simple connected graph G is the length of the shortest path between v a and v b . Vertices v a , v b of G are considered to be resolved by a vertex v if d ( v a , v ) ≠ d ( v b , v ) . An ordered set W = { v 1 , v 2 , v 3 , … , v s } ⊆ V ( G ) is said to be a resolving set for G , if for any v a , v b ∈ V ( G ) , ∃ v i ∈ W ∋ d ( v a , v i ) ≠ d ( v b , v i ) . The representation of vertex v with respect to W is denoted by r ( v | W ) and is an s -vector( s -tuple) ( d ( v , v 1 ) , d ( v , v 2 ) , d ( v , v 3 ) , … , d ( v , v s ) ) . Using representation r ( v | W ) , we can say that W is a resolving set if, for any two vertices v a , v b ∈ V ( G ) , we have r ( v a | W ) ≠ r ( v b | W ) . A minimal resolving set is termed a metric basis for G . The cardinality of the metric basis set is called the metric dimension of G , represented by d i m ( G ) . In this article, we study the metric dimension of two types of bicyclic graphs. The obtained results prove that they have constant metric dimension.

Suggested Citation

  • Asad Khan & Ghulam Haidar & Naeem Abbas & Murad Ul Islam Khan & Azmat Ullah Khan Niazi & Asad Ul Islam Khan, 2023. "Metric Dimensions of Bicyclic Graphs," Mathematics, MDPI, vol. 11(4), pages 1-17, February.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:4:p:869-:d:1062187
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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. 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.
    3. Sedlar, Jelena & Škrekovski, Riste, 2021. "Extremal mixed metric dimension with respect to the cyclomatic number," Applied Mathematics and Computation, Elsevier, vol. 404(C).
    4. 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.
    5. Martin Knor & Jelena Sedlar & Riste Škrekovski, 2022. "Remarks on the Vertex and the Edge Metric Dimension of 2-Connected Graphs," Mathematics, MDPI, vol. 10(14), pages 1-16, July.
    6. Zehui Shao & S. M. Sheikholeslami & Pu Wu & Jia-Biao Liu, 2018. "The Metric Dimension of Some Generalized Petersen Graphs," Discrete Dynamics in Nature and Society, Hindawi, vol. 2018, pages 1-10, August.
    7. Lihua You & Jieshan Yang & Yingxue Zhu & Zhifu You, 2014. "The Maximal Total Irregularity of Bicyclic Graphs," Journal of Applied Mathematics, Hindawi, vol. 2014, pages 1-9, April.
    8. Sedlar, Jelena & Škrekovski, Riste, 2021. "Bounds on metric dimensions of graphs with edge disjoint cycles," Applied Mathematics and Computation, Elsevier, vol. 396(C).
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