Complexity at treewidth three and intermediate grid heights

Determine the complexity of the Partially Ordered Hamiltonian Path Problem on graphs of treewidth 3 and on grid graphs of heights 4, 5, and 6.

Background

The paper establishes completeness for the Partially Ordered Hamiltonian Path Problem (POHPP) at treewidth 4, while providing polynomial-time algorithms at treewidth 2; the treewidth-3 case therefore lies between the tractable and hard regimes. It also proves completeness for grid graphs of height at least 7, leaving heights 4, 5, and 6 unresolved. The authors identify both parameter ranges as the main unresolved consequences of their treewidth- and grid-based complexity results.

References

The complexity results presented in \cref{sec:treewidth} leave two main open questions. What is the complexity of POHPP for graphs of {treewidth}~3 and for grid graphs of height $h \in {4,5,6}$.

— A Graph Width Perspective on Partially Ordered Hamiltonian Paths  (2503.03553 - Beisegel et al., 5 Mar 2025) in Section Open Problems