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Remarks on the Vertex and the Edge Metric Dimension of 2-Connected Graphs

Author

Listed:
  • Martin Knor

    (Faculty of Civil Engineering, Slovak University of Technology in Bratislava, Radlinského 11, 813 68 Bratislava, Slovakia
    These authors contributed equally to this work.)

  • Jelena Sedlar

    (Faculty of Civil Engineering, Architecture and Geodesy, University of Split, 21000 Split, Croatia
    Faculty of Information Studies, 8000 Novo Mesto, Slovenia
    These authors contributed equally to this work.)

  • Riste Škrekovski

    (Faculty of Information Studies, 8000 Novo Mesto, Slovenia
    Faculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana, Slovenia
    These authors contributed equally to this work.)

Abstract

The vertex (respectively edge) metric dimension of a graph G is the size of a smallest vertex set in G , which distinguishes all pairs of vertices (respectively edges) in G , and it is denoted by dim ( G ) (respectively edim ( G ) ). The upper bounds dim ( G ) ≤ 2 c ( G ) − 1 and edim ( G ) ≤ 2 c ( G ) − 1 , where c ( G ) denotes the cyclomatic number of G , were established to hold for cacti without leaves distinct from cycles, and moreover, all leafless cacti that attain the bounds were characterized. It was further conjectured that the same bounds hold for general connected graphs without leaves, and this conjecture was supported by showing that the problem reduces to 2-connected graphs. In this paper, we focus on Θ -graphs, as the most simple 2-connected graphs distinct from the cycle, and show that the the upper bound 2 c ( G ) − 1 holds for both metric dimensions of Θ -graphs; we characterize all Θ -graphs for which the bound is attained. We conclude by conjecturing that there are no other extremal graphs for the bound 2 c ( G ) − 1 in the class of leafless graphs besides already known extremal cacti and extremal Θ -graphs mentioned here.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:14:p:2411-:d:859602
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    References listed on IDEAS

    as
    1. Sedlar, Jelena & Škrekovski, Riste, 2021. "Extremal mixed metric dimension with respect to the cyclomatic number," Applied Mathematics and Computation, Elsevier, vol. 404(C).
    2. 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.
    3. Sedlar, Jelena & Škrekovski, Riste, 2022. "Metric dimensions vs. cyclomatic number of graphs with minimum degree at least two," Applied Mathematics and Computation, Elsevier, vol. 427(C).
    4. Yuezhong Zhang & Suogang Gao, 2020. "On the edge metric dimension of convex polytopes and its related graphs," Journal of Combinatorial Optimization, Springer, vol. 39(2), pages 334-350, February.
    5. 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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    Cited by:

    1. 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.
    2. Sedlar, Jelena & Škrekovski, Riste, 2022. "Metric dimensions vs. cyclomatic number of graphs with minimum degree at least two," Applied Mathematics and Computation, Elsevier, vol. 427(C).
    3. Enqiang Zhu & Shaoxiang Peng & Chanjuan Liu, 2022. "Identifying the Exact Value of the Metric Dimension and Edge Dimension of Unicyclic Graphs," Mathematics, MDPI, vol. 10(19), pages 1-14, September.

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