Advertisement

On the edge metric dimension of convex polytopes and its related graphs

  • 41 Accesses

Abstract

Let \(G=(V, E)\) be a connected graph. The distance between the edge \(e=uv\in E\) and the vertex \(x\in V\) is given by \(d(e, x) = \min \{d(u, x), d(v, x)\}\). A subset \(S_{E}\) of vertices is called an edge metric generator for G if for every two distinct edges \(e_{1}, e_{2}\in E\), there exists a vertex \(x\in S_{E}\) such that \(d(e_{1}, x)\ne d(e_{2}, x)\). An edge metric generator containing a minimum number of vertices is called an edge metric basis for G and the cardinality of an edge metric basis is called the edge metric dimension denoted by \(\mu _{E}(G)\). In this paper, we study the edge metric dimension of some classes of plane graphs. It is shown that the edge metric dimension of convex polytope antiprism \(A_{n}\), the web graph \({\mathbb {W}}_{n}\), and convex polytope \({\mathbb {D}}_{n}\) are bounded, while the prism related graph \(D^{*}_{n}\) has unbounded edge metric dimension.

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

Access options

Buy single article

Instant unlimited access to the full article PDF.

US$ 39.95

Price includes VAT for USA

Subscribe to journal

Immediate online access to all issues from 2019. Subscription will auto renew annually.

US$ 99

This is the net price. Taxes to be calculated in checkout.

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

References

  1. Ali M, Rahim MT, Ali G (2012) On two families of graphs with constant metric dimension. J Prime Res Math 8:95–101

  2. Bača M (1988) Labellings of two classes of convex polytopes. Util Math 34:24–31

  3. Cáceres J, Hernando MC, Mora M, Pelayo IM, Puertas ML, Seara C, Wood DR (2007) On the metric dimension of cartesian products of graphs. SIAM J Discret Math 21(2):423–441

  4. Chartrand G, Zhang P (2003) The theory and applications of resolvability in graphs: a survey. Congr Numer 160:47–68

  5. Chartrand G, Eroh L, Johnson MA, Oellermann OR (2000) Resolvability in graphs and the metric dimension of a graph. Discrete Appl Math 105:99–113

  6. Guo J, Wang K, Li F (2012) Metric dimension of some distance-regular graphs. J Comb Optim 26(1):190–197

  7. Hallaway M, Kang CX, Yi E (2014) On metric dimension of permutation graphs. J Comb Optim 28(4):814–826

  8. Harary F, Melter RA (1976) On the metric dimension of a graph. Ars Comb 2:191–195

  9. Imran M, Baig AQ, Ahmad A (2012) Families of plane graphs with constant metric dimension. Util Math 88:43–57

  10. Imran M, Baig AQ, Bokhary SA (2016) On the metric dimension of rotationally-symmetric graphs. Ars Comb 124:111–128

  11. Javaid I, Rahim MT, Ali K (2008) Families of regular graphs with constant metric dimension. Util Math 75:21–33

  12. Kelenc A, Tratnik N, Yero IG (2018) Uniquely identifying the edges of a graph: the edge metric dimension. Discrete Appl Math. https://doi.org/10.1016/j.dam.2018.05.052

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

  14. Koh KM, Rogers DG, Teo HK, Yap KY (1980) Graceful graphs: some further results and problems. Congr Numer 29:559–571

  15. Kratica J, Filipovic V, Kartelj A (2017) Edge metric dimension of some generalized petersen graphs. arXiv:1807.00580

  16. Peterin I, Yero IG (2018) Edge metric dimension of some graph operations. arXiv:1809.08900v1

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

  18. Slater PJ (1975) Leaves of trees. Congr Numer 14:549–559

  19. Zhu E, Taranenko A, Shao Z, Xu J (2019) On graphs with the maximum edge metric dimension. Discrete Appl Math 257:317–324

  20. Zubrilina N (2018) On the edge dimension of a graph. Discrete Math 341:2083–2088

Download references

Acknowledgements

The authors would like to thank the reviewers for a careful reading of the paper and for many constructive comments. This research is supported by the NSF of China (No.11471097 and No.11971146), the NSF of Hebei Province (No.A2017403010).

Author information

Correspondence to Suogang Gao.

Additional information

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

Verify currency and authenticity via CrossMark

Cite this article

Zhang, Y., Gao, S. On the edge metric dimension of convex polytopes and its related graphs. J Comb Optim 39, 334–350 (2020) doi:10.1007/s10878-019-00472-4

Download citation

Keywords

  • Metric dimension
  • Edge metric dimension
  • Edge metric generator
  • Convex polytopes