Characterization of graphs with basis forced vertices

Characterize, at least partially, all connected graphs whose number of basis forced vertices is nonzero, with particular emphasis on the cases of exactly one and exactly two basis forced vertices.

Background

The paper defines a basis forced vertex as a vertex belonging to every metric basis of a graph and denotes their number by $bf(G)$. It establishes sharp upper bounds for $bf(G)$ and completely characterizes the case $bf(G)=2$ together with metric dimension $n-4$, but it does not provide a general characterization of graphs having at least one basis forced vertex.

The authors explicitly identify the cases $bf(G)=1$ and $bf(G)=2$ as especially relevant. The $bf(G)=2$ case is resolved only under the additional extremal condition $\dim(G)=n-4$, leaving the broader classification problem open.