On the Maximum Number of Vertices that Belong to Every Metric Basis

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 $\mathrm{bf}(G)$. We show that $\mathrm{bf}(G)\le 2/3(n-k-1)$ 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 $\mathrm{bf}(G)\le 2/5(n-1)$ 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 a complete realization of the parameters $n$, $\dim(G)$ and $\mathrm{bf}(G) \ge 1$ within the previous bounds. We consider some extremal cases related to basis forced vertices in a graph, in particular, we give a full characterization of the graphs with $\mathrm{bf}(G) = 2$ and $\dim(G) = n-4$.