Skip to main content
Log in

On the Metric Dimension of Certain Metric Manifolds

  • Original Paper
  • Published:
Bulletin of the Iranian Mathematical Society Aims and scope Submit manuscript

Abstract

In this paper, we find a sharp upper and lower bound for the metric dimension of a geometric manifold with boundary. We also determine the cases in which the bounds hold with equality

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

Access this article

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Bau, S., Lei, Y.: Bisectors in vector groups over integers. Bull. Aust. Math. Soc. 100, 353–361 (2019)

    Article  MathSciNet  Google Scholar 

  2. Bau, S., Beardon, A.F.: The metric dimension of metric spaces. Comput. Methods Funct. Theory 13, 295–305 (2013)

    Article  MathSciNet  Google Scholar 

  3. Barragán-Ramírez, G.A., Estrada-Moreno, A., Ramírez-Cruz, Y., Rodríguez-Velázquez, J.A.: The local metric dimension of the lexicographic product of graphs. Bull. Malays. Math. Sci. Soc. 42(5), 2481–2496 (2019)

    Article  MathSciNet  Google Scholar 

  4. Beardon, A.F.: The Geometry of Discrete Groups. Springer, New York (1983)

    Book  Google Scholar 

  5. Beardon, A.F., Rodríguez-Velázquez, J.A.: On the k-metric dimension of metric spaces. Ars Math. Contemp. 16, 25–38 (2019)

    Article  MathSciNet  Google Scholar 

  6. Blumenthal, L.M.: Theory and Applications of Distance Geometry. Clarendon Press, Oxford (1953)

    MATH  Google Scholar 

  7. Boutin, D.L.: Determining sets, resolving sets, and the exchange property. Graphs Comb. 25, 789–806 (2009)

    Article  MathSciNet  Google Scholar 

  8. Cáceres, J., Hernando, C., Mora, M., Pelayo, I.M., Puertas, M.L.: On the metric dimension of infinite graphs. Electron. Notes Discrete Math. 35, 15–20 (2009)

    Article  MathSciNet  Google Scholar 

  9. Chappell, G.C., Gimbel, J., Hartman, C.: Bounds on the metric and partition dimension of a graph. Ars Comb. 88, 349–366 (2008)

    MathSciNet  MATH  Google Scholar 

  10. Gehér, G.P.: Is it possible to determine a point lying in a simplex if we know the distances from the vertices? J. Math. Anal. Appl. 439, 651–663 (2016)

    Article  MathSciNet  Google Scholar 

  11. Harary, F., Melter, R.A.: On the metric dimension of a graph. ARS Comb. 2, 191–195 (1976)

    MATH  Google Scholar 

  12. Heydarpour, M.: On metric orbit spaces and metric dimension. Topol. Appl. 214, 94–99 (2016)

    Article  MathSciNet  Google Scholar 

  13. Heydarpour, M., Maghsoudi, S.: The metric dimension of geometic spaces. Topol. Appl. 178, 230–235 (2014)

    Article  Google Scholar 

  14. Heydarpour, M., Maghsoudi, S.: The metric dimension of metric manifolds. Bull. Aust. Math. Soc. 91, 508–513 (2015)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  16. Lee, J.M.: Introduction to Smooth Manifolds. Graduate Texts in Mathematics, vol 2018, 2nd edn. Springer, New York (2013)

    Google Scholar 

  17. Matsuzaki, K., Taniguchi, M.: Hyperbolic Manifolds and Kleinian Groups. Oxford University Press, Oxford (1998)

    MATH  Google Scholar 

  18. Melter, R.A., Tomescu, I.: Metric basis in digital geometry. Comput. Vis. Graph. Image Process 25, 113–121 (1984)

    Article  Google Scholar 

  19. Ratcliffe, J.G.: Foundations of Hyperbolic Manifolds. Springer, New York (1994)

    Book  Google Scholar 

  20. Rodríguez-Velázquez, J.A.: Lexicographic metric spaces: basic properties and the metric dimension. Appl. Anal. Discrete Math. (To appear)

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

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Majid Heydarpour.

Additional information

Communicated by Mohammad Reza Koushesh.

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Heydarpour, M., Maghsoudi, S. On the Metric Dimension of Certain Metric Manifolds. Bull. Iran. Math. Soc. 47, 649–657 (2021). https://doi.org/10.1007/s41980-020-00404-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s41980-020-00404-7

Keywords

Mathematics Subject Classification

Navigation