Skip to main content
SpringerLink
Log in

On Graphs of Order n with Metric Dimension \(n-4\)

  • Original Paper
  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

For a connected graph G, let n(G), d(G) and dim(G) denote the order, the diameter and the metric dimension of G, respectively. The twin graph \(G_\text {T}\) is obtained from G by contracting each maximal set of vertices with the same open or close neighborhood into a vertex, respectively. In this paper, we propose a necessary and sufficient condition obtaining G from \(G_\text {T}\) and characterize all graphs with dim\((G)=n-4\) and \(n(G_\text {T})=4\) by this condition. For the graphs with dim\((G)=n-4\), we show the following results: (a) \(4\le n(G_\text {T})\le 7\) if \(d(G)=3\) and the bounds are sharp; (b) \(4\le n(G_\text {T})\le 9\) if \(d(G)=2\).

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
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

Data availability

Not applicable.

Code Availability

Not applicable.

References

  1. Bailey, R.F.: On the metric dimension of incidence graphs. Discrete Math. 341, 1613–1619 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  2. Barrágan-Ramírez, G.A., Rodríguez-Velázquez, J.A.: The local metric dimension of strong product graphs. Graphs Combin. 32, 1263–1278 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Beaudou, L., Dankelmann, P., Foucaud, F., Henning, M.A., Mary, A., Parreau, A.: Bounding the order of a graph using its diameter and metric dimension: a study through tree decompositions and VC dimension. SIAM J. Discrete Math. 32, 902–918 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  4. Beerliova, Z., Eberhard, F., Erlebach, T., Hall, A., Hoffman, M., Mihalak, M., Ram, L.: Network discovery and verification. IEEE J. Sel. Areas Commun. 24, 2168–2181 (2006)

    Article  Google Scholar 

  5. Chartrand, G., Eroh, L., Johnson, M.A., Oellermannn, O.R.: Resolvability in graphs and the metric dimension of a graph. Discrete Appl. Math. 105, 99–113 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chartrand, G., Poisson, C., Zhang, P.: Resolvability and the upper dimension of graphs. Comput. Math. Appl. 39, 19–28 (2000)

    MathSciNet  MATH  Google Scholar 

  7. Chvátal, V.: Mastermind. Combinatorica 3, 325–329 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  8. DasGupta, B., Mobasheri, N.: On optimal approximability results for computing the strong metric dimension. Discrete Appl. Math. 221, 18–24 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. Fernau, H., Rodríguez-Velázquez, J.: 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  MATH  Google Scholar 

  11. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-completeness. Freeman, New York (1979)

    MATH  Google Scholar 

  12. Garijo, D., González, A., Márquez, A.: On the metric dimension, the upper dimension and the resolving number of graphs. Discrete Appl. Math. 161, 1440–1447 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  14. Hernando, C., Mora, M., Pelayo, I.M., Seara, C., Wood, D.R.: Extremal graph theory for metric dimension and diameter. Electron. J. Combin. 17, 30 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Imran, M., Bokhary, S.A.H., Ahmad, A., Semaničová-Feňovčiková, A.: On classes of regular graphs with constant metric dimension. Acta Math. Sci. 33, 187–206 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Jannesari, M., Omoomi, B.: Characterization of \(n\)-vertex graphs with metric dimension \(n-3\). Math. Bohem. 139, 1–23 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kelenc, D., Kuziak, D., Taranenko, A., Yero, I.G.: Mixed metric dimension of graphs. Appl. Math. Comput. 314, 429–438 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Laihonen, T.: The metric dimension for resolving several objects. Inf. Process. Lett. 116, 694–700 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  20. Munir, M., Nizami, A.R., Saeed, H.: On the metric dimension of Möbius Ladder. Ars Combin. 135, 249–256 (2017)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  22. Shanmukha, B., Sooryanarayana, B., Harinath, K.S.: Metric dimension of wheels. Far East J. Appl. Math. 8, 217–229 (2002)

    MathSciNet  MATH  Google Scholar 

  23. Slater, P.J.: Leaves of trees. Congr. Numer. 14, 549–559 (1975)

    MathSciNet  MATH  Google Scholar 

  24. Wang, J., Miao, L., Liu, Y.: Characterization of n-Vertex graphs of metric dimension \(n-3\) by metric matrix. Mathematics 7, 479 (2019)

    Article  Google Scholar 

  25. Wong, D., Zhang, Y., Wang, Z.: Graphs of order \(n\) with determining number \(n-3\). Graphs Combin. 37, 1179–1189 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  26. Yero, I.G., Kuziak, D., Rodríguez-Velázquez, J.A.: On the metric dimension of corona product graphs. Comput. Math. Appl. 61, 2793–2798 (2011)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors thank the editors and anonymous referees for their helpful comments and suggestions, which makes our paper more readable.

Author information

Authors and Affiliations

  1. School of Management, Qufu Normal University, Rizhao, 276826, People’s Republic of China

    Juan Wang & Fenglei Tian

  2. College of Engineering, Qufu Normal University, Rizhao, 276826, People’s Republic of China

    Yunlong Liu

  3. School of Mathematics, China University of Mining and Technology, Xuzhou, 221116, People’s Republic of China

    Jingru Pang & Lianying Miao

Authors
  1. Juan Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fenglei Tian
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yunlong Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jingru Pang
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Lianying Miao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Juan Wang.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest in this paper.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work were partially supported by Shandong Province Natural Science Foundation (Nos. ZR2021MA103, ZR2019BA016), National Natural Science Foundation of China (Nos. 12101354, 11771443), and Rizhao Natural Science Excellent Youth Foundation (No. RZ2021ZR05).

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, J., Tian, F., Liu, Y. et al. On Graphs of Order n with Metric Dimension \(n-4\). Graphs and Combinatorics 39, 29 (2023). https://doi.org/10.1007/s00373-023-02627-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00373-023-02627-x

Keywords

Mathematics Subject Classification

Search

Navigation