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Graphs with the edge metric dimension smaller than the metric dimension

Author

Listed:
  • Knor, Martin
  • Majstorović, Snježana
  • Masa Toshi, Aoden Teo
  • Škrekovski, Riste
  • Yero, Ismael G.

Abstract

Given a connected graph G, the metric (resp. edge metric) dimension of G is the cardinality of the smallest ordered set of vertices that uniquely identifies every pair of distinct vertices (resp. edges) of G by means of distance vectors to such a set. In this work, we settle three open problems on (edge) metric dimension of graphs. Specifically, we show that for every r,t≥2 with r≠t, there is n0, such that for every n≥n0 there exists a graph G of order n with metric dimension r and edge metric dimension t, which among other consequences, shows the existence of infinitely many graph whose edge metric dimension is strictly smaller than its metric dimension. In addition, we also prove that it is not possible to bound the edge metric dimension of a graph G by some constant factor of the metric dimension of G.

Suggested Citation

  • 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).
    • Handle: RePEc:eee:apmaco:v:401:y:2021:i:c:s0096300321001247
    DOI: 10.1016/j.amc.2021.126076
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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.
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    Cited by:

    1. 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.
    2. 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.
    3. 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).
    4. Sedlar, Jelena & Škrekovski, Riste, 2021. "Extremal mixed metric dimension with respect to the cyclomatic number," Applied Mathematics and Computation, Elsevier, vol. 404(C).

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