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Bounds on metric dimensions of graphs with edge disjoint cycles

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  • Sedlar, Jelena
  • Škrekovski, Riste

Abstract

In a graph G, cardinality of the smallest ordered set of vertices that distinguishes every element of V(G) is the (vertex) metric dimension of G. Similarly, the cardinality of such a set is the edge metric dimension of G, if it distinguishes E(G). In this paper these invariants are considered first for unicyclic graphs, and it is shown that the vertex and edge metric dimensions obtain values from two particular consecutive integers, which can be determined from the structure of the graph. In particular, as a consequence, we obtain that these two invariants can differ by at most one for a same unicyclic graph. Next we extend the results to graphs with edge disjoint cycles (i.e. cactus graphs) showing that the two invariants can differ by at most c, where c is the number of cycles in such a graph. We conclude the paper with a conjecture that generalizes the previously mentioned consequences to graphs with prescribed cyclomatic number c by claiming that the difference of the invariant is still bounded by c.

Suggested Citation

  • Sedlar, Jelena & Škrekovski, Riste, 2021. "Bounds on metric dimensions of graphs with edge disjoint cycles," Applied Mathematics and Computation, Elsevier, vol. 396(C).
    • Handle: RePEc:eee:apmaco:v:396:y:2021:i:c:s0096300320308614
    DOI: 10.1016/j.amc.2020.125908
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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. 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. 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. Sedlar, Jelena & Škrekovski, Riste, 2021. "Extremal mixed metric dimension with respect to the cyclomatic number," Applied Mathematics and Computation, Elsevier, vol. 404(C).
    2. 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.
    3. 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.
    4. 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).
    5. 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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