Skip to main content
SpringerLink
Log in

The Local Metric Dimension of the Lexicographic Product of Graphs

  • Published:
Bulletin of the Malaysian Mathematical Sciences Society Aims and scope Submit manuscript

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 \(\mathcal {H}\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Notes

  1. Adjacency generators were called adjacency resolving sets in [16].

References

  1. Barragán-Ramírez, G.A., Rodríguez-Velázquez, J.A.: The local metric dimension of strong product graphs. Gr. Comb. 32(4), 1263–1278 (2016)

    Article  MathSciNet  Google Scholar 

  2. Blumenthal, L.M.: Theory and Applications of Distance Geometry, 2nd edn. Chelsea Publishing Co., New York (1970)

    MATH  Google Scholar 

  3. Brigham, R.C., Chartrand, G., Dutton, R.D., Zhang, P.: Resolving domination in graphs. Mathematica Bohemica 128(1), 25–36 (2003)

    MathSciNet  MATH  Google Scholar 

  4. Chartrand, G., Saenpholphat, V., Zhang, P.: The independent resolving number of a graph. Mathematica Bohemica 128(4), 379–393 (2003)

    MathSciNet  MATH  Google Scholar 

  5. Estrada-Moreno, A., García-Gómez, C., Ramírez-Cruz, Y., Rodríguez-Velázquez, J.: The simultaneous strong metric dimension of graph families. Bull. Malays. Math. Sci. Soc. 39(1), 175–192 (2016)

    Article  MathSciNet  Google Scholar 

  6. Estrada-Moreno, A., Ramírez-Cruz, Y., Rodríguez-Velázquez, J.A.: On the adjacency dimension of graphs. Appl. Anal. Discrete Math. 10, 102–127 (2016)

    Article  MathSciNet  Google Scholar 

  7. Estrada-Moreno, A., Rodríguez-Velázquez, J.A., Yero, I.G.: The \(k\)-metric dimension of a graph. Appl. Math. Inf. Sci. 9(6), 2829–2840 (2015)

    MathSciNet  Google Scholar 

  8. Estrada-Moreno, A., Yero, I., Rodríguez-Velázquez, J.: The \(k\)-metric dimension of the lexicographic product of graphs. Discrete Math. 339(7), 1924–1934 (2016)

    Article  MathSciNet  Google Scholar 

  9. Feng, M., Wang, K.: On the fractional metric dimension of corona product graphs and lexicographic product graphs. arXiv:1206.1906 [math.CO]

  10. Fernau, H., Rodríguez-Velázquez, J. A.: On the (adjacency) metric dimension of corona and strong product graphs and their local variants: combinatorial and computational results. Discrete Appl. Math. 236, 183–202 (2018)

    Article  MathSciNet  Google Scholar 

  11. Fernau, H., Rodríguez-Velázquez, J. A.: Notions of metric dimension of corona products: combinatorial and computational results. In: Hirsch, E.A., Kuznetsov, S.O., Pin, J.-É., Vereshchagin, N.K. (eds.) Computer Science-Theory and Applications. Lecture Notes in Comput. Sci., vol. 8476, pp. 153–166. Springer, Cham (2014).

    MATH  Google Scholar 

  12. Hammack, R., Imrich, W., Klavžar, S.: Handbook of Product Graphs, Discrete Mathematics and its Applications, 2nd edn. CRC Press, Boca Raton (2011)

    Book  Google Scholar 

  13. Harary, F.: Graph Theory. Addison-Wesley Publishing Co., Reading, Mass.-Menlo Park, Calif.-London (1969)

    Book  Google Scholar 

  14. Harary, F., Melter, R.A.: On the metric dimension of a graph. Ars Combinatoria 2, 191–195 (1976)

    MathSciNet  MATH  Google Scholar 

  15. Imrich, W., Klavžar, S.: Product Graphs, Structure and Recognition. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, New York (2000)

    MATH  Google Scholar 

  16. Jannesari, M., Omoomi, B.: The metric dimension of the lexicographic product of graphs. Discrete Math. 312(22), 3349–3356 (2012)

    Article  MathSciNet  Google Scholar 

  17. Johnson, M.: Structure–activity maps for visualizing the graph variables arising in drug design. J. Biopharm. Stat. 3(2), 203–236 (1993). (pMID: 8220404)

    Article  MathSciNet  Google Scholar 

  18. Johnson, M.: Browsable structure–activity datasets. In: Carbó-Dorca, R., Mezey, P. (eds.) Advances in Molecular Similarity. chap. 8, pp. 153–170. JAI Press Inc, Stamford (1998)

    Google Scholar 

  19. Khuller, S., Raghavachari, B., Rosenfeld, A.: Landmarks in graphs. Discrete Appl. Math. 70(3), 217–229 (1996)

    Article  MathSciNet  Google Scholar 

  20. Kuziak, D., Yero, I.G., Rodríguez-Velázquez, J.A.: Closed formulae for the strong metric dimension of lexicographic product graphs. Discussiones Mathematicae Graph Theory 36(4), 1051–1064 (2016)

    Article  MathSciNet  Google Scholar 

  21. Okamoto, F., Phinezy, B., Zhang, P.: The local metric dimension of a graph. Mathematica Bohemica 135(3), 239–255 (2010)

    MathSciNet  MATH  Google Scholar 

  22. Ramírez-Cruz, Y., Estrada-Moreno, A., Rodríguez-Velázquez, J.A.: The simultaneous metric dimension of families composed by lexicographic product graphs. Gr. Combin. 32(5), 2093–2120 (2016)

    Article  MathSciNet  Google Scholar 

  23. Ramírez-Cruz, Y., Oellermann, O.R., Rodríguez-Velázquez, J.A.: The simultaneous metric dimension of graph families. Discrete Appl. Math. 198, 241–250 (2016)

    Article  MathSciNet  Google Scholar 

  24. Rodríguez-Velázquez, J.A., Barragán-Ramírez, G.A., García-Gómez, C.: On the local metric dimension of corona product graphs. Bull. Malays. Math. Sci. Soc. 39(1), 157–173 (2016)

    Article  MathSciNet  Google Scholar 

  25. Rodríguez-Velázquez, J.A., García-Gómez, C., Barragán-Ramírez, G.A.: Computing the local metric dimension of a graph from the local metric dimension of primary subgraphs. Int. J. Comput. Math. 92(4), 686–693 (2015)

    Article  MathSciNet  Google Scholar 

  26. Saputro, S., Simanjuntak, R., Uttunggadewa, S., Assiyatun, H., Baskoro, E., Salman, A., Bača, M.: The metric dimension of the lexicographic product of graphs. Discrete Math. 313(9), 1045–1051 (2013)

    Article  MathSciNet  Google Scholar 

  27. Sebö, A., Tannier, E.: On metric generators of graphs. Math. Oper. Res. 29(2), 383–393 (2004)

    Article  MathSciNet  Google Scholar 

  28. Slater, P.J.: Leaves of trees. Congressus Numerantium 14, 549–559 (1975)

    MathSciNet  MATH  Google Scholar 

  29. Zykov, A.A.: On some properties of linear complexes. Matematičeskiǐ Sbornik (N.S.) 24(66), 163–188 (1949)

    MathSciNet  Google Scholar 

Download references

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).

Author information

Authors and Affiliations

  1. Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, Av. Països Catalans 26, 43007, Tarragona, Spain

    Gabriel A. Barragán-Ramírez & Juan A. Rodríguez-Velázquez

  2. IN3 – Computer Science Department, Open University of Catalonia, Av. Carl Friedrich Gauss 5, 08860, Castelldefels, Spain

    Alejandro Estrada-Moreno

  3. Interdisciplinary Centre for Security, Reliability and Trust, University of Luxembourg, 6 av. de la Fonte, 4364, Esch-sur-Alzette, Luxembourg

    Yunior Ramírez-Cruz

Authors
  1. Gabriel A. Barragán-Ramírez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alejandro Estrada-Moreno
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yunior Ramírez-Cruz
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Juan A. Rodríguez-Velázquez
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yunior Ramírez-Cruz.

Additional information

Communicated by Sanming Zhou.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Barragán-Ramírez, G.A., Estrada-Moreno, A., Ramírez-Cruz, Y. et al. The Local Metric Dimension of the Lexicographic Product of Graphs. Bull. Malays. Math. Sci. Soc. 42, 2481–2496 (2019). https://doi.org/10.1007/s40840-018-0611-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40840-018-0611-3

Keywords

Mathematics Subject Classification

Search

Navigation