On a Tight Bound for the Maximum Number of Vertices that Belong to Every Metric Basis
: Hakanen, Anni; Junnila, Ville; Laihonen, Tero; Miikonen, Havu; Yero, Ismael G.
: Gaur, Daya; Mathew, Rogers
: Conference on Algorithms and Discrete Applied Mathematics
Publisher: Springer Nature Switzerland
: 2025
: Lecture Notes in Computer Science
: Algorithms and Discrete Applied Mathematics: 11th International Conference, CALDAM 2025, Coimbatore, India, February 13–15, 2025, Proceedings
: Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
: 15536
: 173
: 184
: 978-3-031-83437-0
: 978-3-031-83438-7
: 0302-9743
: 1611-3349
DOI: https://doi.org/10.1007/978-3-031-83438-7_15
: https://doi.org/10.1007/978-3-031-83438-7_15
: https://research.utu.fi/converis/portal/detail/Publication/491723438
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 bf(G). We show that bf(G)≤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 bf(G)≤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.
:
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).
