Classification by metric dimension two and fault-tolerant metric dimension

Characterize all maximal outerplanar graphs with metric dimension 2 and fault-tolerant metric dimension equal to each fixed value $k\in\{4,5,6\}$.

Background

For maximal outerplanar graphs of order at least 7 and metric dimension 2, the paper proves that the fault-tolerant metric dimension lies between 4 and 6 and shows that all values permitted by the bounds are realizable in appropriate examples. A complete structural characterization of the graphs attaining each value 4, 5, or 6 remains unresolved.

References

Characterize all MOPs $G$ such that $\beta (G)=2$ and $\beta'(G)=k$, for each fixed value of $k\in {4,5,6}$.

— On the fault-tolerant metric dimension of maximal outerplanar graphs  (2610.13374 - Hernando et al., 9 Oct 2026) in Section Conclusions, second Open problem