A set $S\subseteq V(G)$ is a resolving set in a graph $G$ if for any pair $u,v\in V(G)$ there exists $s\in S$ such that $d(u,s)\neq d(v,s)$. A metric basis is a resolving set of the smallest possible cardinality. It is known that there are graphs where some vertices must belong to every metric basis. We call these vertices basis forced vertices. In this talk, we give, for example, bounds on the size of a graph with $k$ basis forced vertices.
This is a joint work with Anni Hakanen, Ville Junnila and Ismael Yero.