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Extremal mixed metric dimension with respect to the cyclomatic number

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  • Sedlar, Jelena
  • Škrekovski, Riste

Abstract

In a graph G, the cardinality of the smallest ordered set of vertices that distinguishes every element of V(G)∪E(G) is called the mixed metric dimension of G, and it is denoted by mdim(G). In [12] it was conjectured that every graph G with cyclomatic number c(G) satisfies mdim(G)≤L1(G)+2c(G) where L1(G) is the number of leaves in G. It is already proven that the equality holds for all trees and more generally for graphs with edge-disjoint cycles in which every cycle has precisely one vertex of degree ≥3. In this paper we determine that for every Θ-graph G, the mixed metric dimension mdim(G) equals 3 or 4, with 4 being attained if and only if G is a balanced Θ-graph. Thus, for balanced Θ-graphs the above inequality is also tight. We conclude the paper by further conjecturing that there are no other graphs, besides the ones mentioned here, for which the equality mdim(G)=L1(G)+2c(G) holds.

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  • Sedlar, Jelena & Škrekovski, Riste, 2021. "Extremal mixed metric dimension with respect to the cyclomatic number," Applied Mathematics and Computation, Elsevier, vol. 404(C).
    • Handle: RePEc:eee:apmaco:v:404:y:2021:i:c:s0096300321003283
    DOI: 10.1016/j.amc.2021.126238
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    References listed on IDEAS

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    1. András Sebő & Eric Tannier, 2004. "On Metric Generators of Graphs," Mathematics of Operations Research, INFORMS, vol. 29(2), pages 383-393, May.
    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. 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. 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.
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    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. 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.
    3. Nie, Kairui & Xu, Kexiang, 2023. "Mixed metric dimension of some graphs," Applied Mathematics and Computation, Elsevier, vol. 442(C).
    4. 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.
    5. 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).

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