A4 Refereed article in a conference publication
On a Tight Bound for the Maximum Number of Vertices that Belong to Every Metric Basis
Authors: Hakanen, Anni; Junnila, Ville; Laihonen, Tero; Miikonen, Havu; Yero, Ismael G.
Editors: Gaur, Daya; Mathew, Rogers
Conference name: Conference on Algorithms and Discrete Applied Mathematics
Publisher: Springer Nature Switzerland
Publication year: 2025
Journal: Lecture Notes in Computer Science
Book title : Algorithms and Discrete Applied Mathematics: 11th International Conference, CALDAM 2025, Coimbatore, India, February 13–15, 2025, Proceedings
Journal name in source: Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume: 15536
First page : 173
Last page: 184
ISBN: 978-3-031-83437-0
eISBN: 978-3-031-83438-7
ISSN: 0302-9743
eISSN: 1611-3349
DOI: https://doi.org/10.1007/978-3-031-83438-7_15(external)
Web address : https://doi.org/10.1007/978-3-031-83438-7_15(external)
Self-archived copy’s web address: https://research.utu.fi/converis/portal/detail/Publication/491723438(external)
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.
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Funding information in the publication:
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).
