Abstract
Metric bases of graphs have been widely studied since their introduction in the 1970’s by Slater and, independently, by Harary and Melter. In this paper, we concentrate on the existence of vertices in a graph G that belong to all metric bases of G. We call these basis forced vertices, and denote the number of them by \(\textrm{bf}(G)\). We show that \(\textrm{bf}(G)\le 2/3(n-k-1)\) for any connected nontrivial graph G of order n having k vertices in each metric basis. In addition, we show that this bound can be attained. Furthermore, the previous result implies the bound \(\textrm{bf}(G)\le 2/5(n-1)\) formulated in terms of the order n of the graph for any nontrivial connected graph G. This result answers a question posed by Bagheri et al. in 2016. Moreover, we provide some realization results and consider some extremal cases related to basis forced vertices in a graph.
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References
Gh, B.B., Jannesari, M., Omoomi, B.: Unique basis graphs. Ars Comb. 129, 249–259 (2016)
Buczkowski, P.S., Chartrand, G., Poisson, C., Zhang, P.: On k-dimensional graphs and their bases. Period. Math. Hung. 46, 9–15 (2003). https://doi.org/10.1023/A:1025745406160
Chartrand, G., Eroh, L., Johnson, M.A., Oellermann, O.R.: Resolvability in graphs and the metric dimension of a graph. Discrete Appl. Math. 105, 99–113 (2000). https://doi.org/10.1016/S0166-218X(00)00198-0
Claverol, M., et al.: Metric dimension of maximal outerplanar graphs. Bull. Malays. Math. Sci. Soc. 44(4), 2603–2630 (2021). https://doi.org/10.1007/s40840-020-01068-6
Foster-Greenwood, B., Uhl, Ch.: Metric dimension of a direct product of three complete graphs. Electron. J. Combin. 31, Paper No. 2.13 (2024). https://doi.org/10.37236/12399
Hakanen, A., Junnila, V., Laihonen, T.: The solid-metric dimension. Theor. Comput. Sci. 806, 156–170 (2020). https://doi.org/10.1016/j.tcs.2019.02.013
Hakanen, A., Junnila, V., Laihonen, T., Miikonen, H., Yero, I.G.: On the Maximum Number of Vertices that Belong to Every Metric Basis (in preparation)
Hakanen, A., Junnila, V., Laihonen, T., Puertas, M.L.: On the metric dimensions for sets of vertices. Discuss. Math. Graph Theory 43(1), 245–275 (2023). https://doi.org/10.7151/dmgt.2367
Hakanen, A., Junnila, V., Laihonen, T., Yero, I.G.: On vertices contained in all or in no metric basis. Discrete Appl. Math. 319, 407–423 (2022). https://doi.org/10.1016/j.dam.2021.12.004
Hakanen, A., Junnila, V., Laihonen, T., Yero, I.G.: On the unicyclic graphs having vertices that belong to all their (strong) metric bases. Discrete Appl. Math. 353, 191–207 (2024). https://doi.org/10.1016/j.dam.2024.04.020
Harary, F., Melter, R.: On the metric dimension of a graph. Ars Combin. 2, 191–195 (1976)
Hernando, C., Mora, M., Pelayo, I.M.: Locating domination in bipartite graphs and their complements. Discrete Appl. Math. 263, 195–203 (2019). https://doi.org/10.1016/j.dam.2018.09.034
Mashkaria, S., Ódor, G., Thiran, P.: On the robustness of the metric dimension of grid graphs to adding a single edge. Discrete Appl. Math. 316, 1–27 (2022). https://doi.org/10.1016/j.dam.2022.02.014
Sedlar, J., Škrekovski, R.: Bounds on metric dimensions of graphs with edge disjoint cycles. Appl. Math. Comput. 396, 125908 (2021). https://doi.org/10.1016/j.amc.2020.125908
Sedlar, J., Škrekovski, R.: Metric dimensions vs. cyclomatic number of graphs with minimum degree at least two. Appl. Math. Comput. 427, 127147 (2022). https://doi.org/10.1016/j.amc.2022.127147
Sedlar, J., Škrekovski, R.: Vertex and edge metric dimensions of unicyclic graphs. Discrete Appl. Math. 314, 81–92 (2022). https://doi.org/10.1016/j.dam.2022.02.022
Slater, P.J.: Leaves of trees. Congr. Numer. 14, 549–559 (1975)
Tillquist, R.C., Lladser, M.E.: Low-dimensional representation of genomic sequences. J. Math. Biol. 79(1), 1–29 (2019). https://doi.org/10.1007/s00285-019-01348-1
Tillquist, R.C., Frongillo, R.M., Lladser, M.E.: Getting the lay of the land in discrete space: a survey of metric dimension and its applications. SIAM Rev. 65(4), 919–962 (2023). https://doi.org/10.1137/21M1409512
Wang, J., Tian, F., Liu, Y., Pang, J., Miao, L.: On graphs of order \(n\) with metric dimension \(n-4\). Graphs Combin. 39, Paper No. 29 (2023). https://doi.org/10.1007/s00373-023-02627-x
Wu, J., Wang, L., Yang, W.: Learning to compute the metric dimension of graphs. Appl. Math. Comput. 432, 127350 (2022). https://doi.org/10.1016/j.amc.2022.127350
Acknowledgments
Ismael G. Yero has been partially supported by the Spanish Ministry of Science and Innovation through the grant PID2023-146643NB-I00. Ville Junnila, Tero Laihonen and Havu Miikonen have been partially supported by Academy of Finland grant number 338797. Anni Hakanen was supported by Turku Collegium for Science, Medicine and Technology (TCSMT).
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Hakanen, A., Junnila, V., Laihonen, T., Miikonen, H., Yero, I. (2025). On a Tight Bound for the Maximum Number of Vertices that Belong to Every Metric Basis. In: Gaur, D., Mathew, R. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2025. Lecture Notes in Computer Science, vol 15536. Springer, Cham. https://doi.org/10.1007/978-3-031-83438-7_15
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