Characterization by metric dimension and number of basis forced vertices

Characterize graphs according to prescribed combinations of metric dimension and number of basis forced vertices, extending the extremal characterization established for graphs with $bf(G)=2$ and $\dim(G)=n-4$.

Background

The paper completely realizes all numerically admissible triples (n,dim⁡(G),bf(G))(n,\dim(G),bf(G)) within the bounds it proves, and it characterizes one extremal family with exactly two basis forced vertices and metric dimension n−4n-4. However, it does not classify the graph structures associated with different metric dimensions and numbers of basis forced vertices.

The proposed problem asks for structural characterizations rather than merely existence results, and is explicitly motivated by Theorem characterizing the case bf(G)=2bf(G)=2 and dim⁡(G)=n−4\dim(G)=n-4.

References

In the light of Theorem~\ref{char2bf}, it would be interesting to characterize graphs with different metric dimensions and number of basis forced vertices.

— On the Maximum Number of Vertices that Belong to Every Metric Basis  (2608.24336 - Hakanen et al., 25 Aug 2026) in Section 6, Concluding remarks

In this sense, a weaker version of the item above concerns finding the graphs $G$ of order $n$ with metric dimension $n-5$ and one or two basis forced vertices.

— On the Maximum Number of Vertices that Belong to Every Metric Basis  (2608.24336 - Hakanen et al., 25 Aug 2026) in Section 6, Concluding remarks