Metric Dimension of Maximal Outerplanar Graphs

Abstract

In this paper, we study the metric dimension problem in maximal outerplanar graphs. Concretely, if \(\beta (G)\) denotes the metric dimension of a maximal outerplanar graph G of order n, we prove that \(2\le \beta (G) \le \lceil \frac{2n}{5}\rceil \) and that the bounds are tight. We also provide linear algorithms to decide whether the metric dimension of G is 2 and to build a resolving set S of size \(\lceil \frac{2n}{5}\rceil \) for G. Moreover, we characterize all maximal outerplanar graphs with metric dimension 2.

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Acknowledgements

The authors would like to thank the anonymous referees for valuable comments which helped to improve the paper. A. García, M. Mora and J. Tejel are supported by H2020-MSCA-RISE project 734922-CONNECT; M. Claverol, A. García, G. Hernández, C. Hernando, M. Mora and J. Tejel are supported by Project MTM2015-63791-R (MINECO/FEDER); M. Claverol, C. Hernando, M. Mora and J. Tejel are supported by project PID2019-104129GB-I00/AEI/10.13039/501100011033 of the Spanish Ministry of Science and Innovation; M. Claverol is supported by project Gen. Cat. DGR 2017SGR1640; C. Hernando, M. Maureso and M. Mora are supported by project Gen. Cat. DGR 2017SGR1336; and A. García and J. Tejel are supported by project Gobierno de Aragón E41-17R (FEDER).

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Correspondence to M. Mora.

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This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 734922.

Communicated by Sandi Klavžar.

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Claverol, M., García, A., Hernández, G. et al. Metric Dimension of Maximal Outerplanar Graphs. Bull. Malays. Math. Sci. Soc. (2021). https://doi.org/10.1007/s40840-020-01068-6

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Keywords

  • Metric dimension
  • Resolving set
  • Maximal outerplanar graph

Mathematics Subject Classification

  • 05C12
  • 05C10