Metric dimension on 2-trees

Determine the computational complexity of Metric Dimension on 2-trees, the maximal graphs of treewidth two.

Background

A 2-tree is a maximal graph of treewidth two. The reduction establishes NP-completeness for graphs of treewidth at most two, but its constructed graphs are far from maximal. Consequently, the complexity of Metric Dimension on the class of 2-trees is left unresolved.

References

The complexity on 2-trees is open.

— Metric Dimension is NP-complete on Graphs of Treewidth Two  (2610.21133 - Benoit, 15 Oct 2026) in Section 1, subsection “Open questions”