Exact pathwidth threshold for NP-completeness

Determine the least pathwidth for which Metric Dimension is NP-complete.

Background

Earlier work cited in the paper gives NP-hard instances of pathwidth at most 24. The present construction proves hardness at treewidth two but does not determine the pathwidth of its graphs. Thus the smallest pathwidth at which NP-completeness occurs remains unresolved.

References

We do not determine the pathwidth of $G_\varphi$ here; the least pathwidth at which the problem is NP-complete is open.

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