IDEAS home Printed from https://ideas.repec.org/a/eee/apmaco/v427y2022ics0096300322002272.html
   My bibliography  Save this article

Metric dimensions vs. cyclomatic number of graphs with minimum degree at least two

Author

Listed:
  • Sedlar, Jelena
  • Škrekovski, Riste

Abstract

The vertex (resp. edge) metric dimension of a connected graph G, denoted by dim(G) (resp. edim(G)), is defined as the size of a smallest set S⊆V(G) which distinguishes all pairs of vertices (resp. edges) in G. Bounds dim(G)≤L(G)+2c(G) and edim(G)≤L(G)+2c(G), where c(G) is the cyclomatic number in G and L(G) depends on the number of leaves in G, are known to hold for cacti and it is conjectured that they hold for general graphs. In leafless graphs it holds that L(G)=0, so for such graphs the conjectured upper bound becomes 2c(G). In this paper, we show that the bound 2c(G) cannot be attained by leafless cacti, so the upper bound for such cacti decreases to 2c(G)−1, and we characterize all extremal leafless cacti for the decreased bound. We conjecture that the decreased bound holds for all leafless graphs, i.e. graphs with minimum degree at least two. We support this conjecture by showing that it holds for all graphs with minimum degree at least three and that it is sufficient to show that it holds for all 2-connected graphs, and we also verify the conjecture for graphs of small order.

Suggested Citation

  • 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).
  • Handle: RePEc:eee:apmaco:v:427:y:2022:i:c:s0096300322002272
    DOI: 10.1016/j.amc.2022.127147
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0096300322002272
    Download Restriction: Full text for ScienceDirect subscribers only

    File URL: https://libkey.io/10.1016/j.amc.2022.127147?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. 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.
    2. Sedlar, Jelena & Škrekovski, Riste, 2021. "Bounds on metric dimensions of graphs with edge disjoint cycles," Applied Mathematics and Computation, Elsevier, vol. 396(C).
    3. 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).
    4. Sedlar, Jelena & Škrekovski, Riste, 2021. "Extremal mixed metric dimension with respect to the cyclomatic number," Applied Mathematics and Computation, Elsevier, vol. 404(C).
    5. 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.
    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. 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.

    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. 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. 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. 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.
    4. Sedlar, Jelena & Škrekovski, Riste, 2021. "Extremal mixed metric dimension with respect to the cyclomatic number," Applied Mathematics and Computation, Elsevier, vol. 404(C).
    5. Nie, Kairui & Xu, Kexiang, 2023. "Mixed metric dimension of some graphs," Applied Mathematics and Computation, Elsevier, vol. 442(C).
    6. 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.
    7. 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.
    8. Sedlar, Jelena & Škrekovski, Riste, 2021. "Bounds on metric dimensions of graphs with edge disjoint cycles," Applied Mathematics and Computation, Elsevier, vol. 396(C).
    9. 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.
    10. 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.
    11. Dorota Kuziak, 2020. "The Strong Resolving Graph and the Strong Metric Dimension of Cactus Graphs," Mathematics, MDPI, vol. 8(8), pages 1-14, August.
    12. Rashad Ismail & Asim Nadeem & Kamran Azhar, 2024. "Local Metric Resolvability of Generalized Petersen Graphs," Mathematics, MDPI, vol. 12(14), pages 1-14, July.

    More about this item

    Statistics

    Access and download statistics

    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:eee:apmaco:v:427:y:2022:i:c:s0096300322002272. 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: Catherine Liu (email available below). General contact details of provider: https://www.journals.elsevier.com/applied-mathematics-and-computation .

    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.