IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v7y2019i5p479-d234532.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/7/5/479/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/7/5/479/
    Download Restriction: no
    ---><---

    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. 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. 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.
    4. 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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Sedlar, Jelena & Škrekovski, Riste, 2021. "Bounds on metric dimensions of graphs with edge disjoint cycles," Applied Mathematics and Computation, Elsevier, vol. 396(C).
    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. Michael Hallaway & Cong X. Kang & Eunjeong Yi, 2014. "On metric dimension of permutation graphs," Journal of Combinatorial Optimization, Springer, vol. 28(4), pages 814-826, November.
    5. 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.
    6. Ismael González Yero, 2020. "The Simultaneous Strong Resolving Graph and the Simultaneous Strong Metric Dimension of Graph Families," Mathematics, MDPI, vol. 8(1), pages 1-11, January.
    7. González, Antonio & Hernando, Carmen & Mora, Mercè, 2018. "Metric-locating-dominating sets of graphs for constructing related subsets of vertices," Applied Mathematics and Computation, Elsevier, vol. 332(C), pages 449-456.
    8. 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).
    9. Knor, Martin & Majstorović, Snježana & Masa Toshi, Aoden Teo & Škrekovski, Riste & Yero, Ismael G., 2021. "Graphs with the edge metric dimension smaller than the metric dimension," Applied Mathematics and Computation, Elsevier, vol. 401(C).
    10. Mladenović, Nenad & Kratica, Jozef & Kovačević-Vujčić, Vera & Čangalović, Mirjana, 2012. "Variable neighborhood search for metric dimension and minimal doubly resolving set problems," European Journal of Operational Research, Elsevier, vol. 220(2), pages 328-337.
    11. Nie, Kairui & Xu, Kexiang, 2023. "Mixed metric dimension of some graphs," Applied Mathematics and Computation, Elsevier, vol. 442(C).
    12. 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).
    13. 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.
    14. Yero, Ismael G. & Estrada-Moreno, Alejandro & Rodríguez-Velázquez, Juan A., 2017. "Computing the k-metric dimension of graphs," Applied Mathematics and Computation, Elsevier, vol. 300(C), pages 60-69.
    15. Sunny Kumar Sharma & Vijay Kumar Bhat, 2022. "On metric dimension of plane graphs with $$\frac{m}{2}$$ m 2 number of 10 sided faces," Journal of Combinatorial Optimization, Springer, vol. 44(3), pages 1433-1458, October.
    16. Jun Guo & Kaishun Wang & Fenggao Li, 2013. "Metric dimension of some distance-regular graphs," Journal of Combinatorial Optimization, Springer, vol. 26(1), pages 190-197, July.
    17. Muhammad Azeem & Muhammad Kamran Jamil & Yilun Shang, 2023. "Notes on the Localization of Generalized Hexagonal Cellular Networks," Mathematics, MDPI, vol. 11(4), pages 1-15, February.
    18. 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.
    19. Dorota Kuziak, 2020. "The Strong Resolving Graph and the Strong Metric Dimension of Cactus Graphs," Mathematics, MDPI, vol. 8(8), pages 1-14, August.
    20. Ron Adar & Leah Epstein, 2017. "The k-metric dimension," Journal of Combinatorial Optimization, Springer, vol. 34(1), pages 1-30, July.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:7:y:2019:i:5:p:479-:d:234532. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.