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The Local Metric Dimension of the Lexicographic Product of Graphs

  • Gabriel A. Barragán-Ramírez
  • Alejandro Estrada-Moreno
  • Yunior Ramírez-Cruz
  • Juan A. Rodríguez-Velázquez
Article

Abstract

The metric dimension is quite a well-studied graph parameter. Recently, the local metric dimension and the adjacency dimension have been introduced and studied. In this paper, we give a general formula for the local metric dimension of the lexicographic product \(G \circ \mathcal {H}\) of a connected graph G of order n and a family \(\mathcal {H}\) composed of n graphs. We show that the local metric dimension of \(G \circ \mathcal {H}\) can be expressed in terms of the numbers of vertices in the true twin equivalence classes of G, and the local adjacency dimension of the graphs in \(\mathcal {H}\).

Keywords

Local metric dimension Local adjacency dimension Lexicographic product graphs 

Mathematics Subject Classification

05C12 05C76 

Notes

Acknowledgements

This work has been partially supported by the Spanish Ministerio de Economía y Competitividad (MTM2016-78227-C2-1-P, TRA2013-48180-C3-P and TRA2015-71883-REDT).

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Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2018

Authors and Affiliations

  • Gabriel A. Barragán-Ramírez
    • 1
  • Alejandro Estrada-Moreno
    • 2
  • Yunior Ramírez-Cruz
    • 3
  • Juan A. Rodríguez-Velázquez
    • 1
  1. 1.Departament d’Enginyeria Informàtica i MatemàtiquesUniversitat Rovira i VirgiliTarragonaSpain
  2. 2.IN3 – Computer Science DepartmentOpen University of CataloniaCastelldefelsSpain
  3. 3.Interdisciplinary Centre for Security, Reliability and TrustUniversity of LuxembourgEsch-sur-AlzetteLuxembourg

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