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Identifying the Exact Value of the Metric Dimension and Edge Dimension of Unicyclic Graphs

Author

Listed:
  • Enqiang Zhu

    (Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, China)

  • Shaoxiang Peng

    (Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, China)

  • Chanjuan Liu

    (School of Computer Science and Technology, Dalian University of Technology, Dalian 116024, China)

Abstract

Given a simple connected graph G , the metric dimension dim ( G ) (and edge metric dimension edim ( G ) ) is defined as the cardinality of a smallest vertex subset S ⊆ V ( G ) for which every two distinct vertices (and edges) in G have distinct distances to a vertex of S . It is an interesting topic to discuss the relation between these two dimensions for some class of graphs. This paper settles two open problems on this topic for unicyclic graphs. We recently learned that Sedlar and Škrekovski settled these problems, but our work presents the results in a completely different way. By introducing four classes of subgraphs, we characterize the structure of a unicyclic graph G such that dim ( G ) and edim ( G ) are equal to the cardinality of any minimum branch-resolving set for unicyclic graphs. This generates an approach to determine the exact value of the metric dimension (and edge metric dimension) for a unicyclic graph.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:19:p:3539-:d:928228
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    References listed on IDEAS

    as
    1. Bo Deng & Muhammad Faisal Nadeem & Muhammad Azeem, 2021. "On the Edge Metric Dimension of Different Families of Möbius Networks," Mathematical Problems in Engineering, Hindawi, vol. 2021, pages 1-9, March.
    2. 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.
    3. Meiqin Wei & Jun Yue & Lily Chen, 2022. "The effect of vertex and edge deletion on the edge metric dimension of graphs," Journal of Combinatorial Optimization, Springer, vol. 44(1), pages 331-342, August.
    4. 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.
    5. Sedlar, Jelena & Škrekovski, Riste, 2021. "Bounds on metric dimensions of graphs with edge disjoint cycles," Applied Mathematics and Computation, Elsevier, vol. 396(C).
    6. David G. L. Wang & Monica M. Y. Wang & Shiqiang Zhang, 2022. "Determining the edge metric dimension of the generalized Petersen graph P(n, 3)," Journal of Combinatorial Optimization, Springer, vol. 43(2), pages 460-496, March.
    7. Tanveer Iqbal & Muhammad Rafiq & Muhammad Naeem Azhar & Muhammad Salman & Imran Khalid & Ali Ahmad, 2022. "On the Edge Resolvability of Double Generalized Petersen Graphs," Journal of Mathematics, Hindawi, vol. 2022, pages 1-14, April.
    Full references (including those not matched with items on IDEAS)

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