Para-hardness for distance to outerplanar and edge clique cover number

Determine whether the Partially Ordered Hamiltonian Path Problem is para-hard when parameterized by distance to outerplanar and when parameterized by edge clique cover number.

Background

The paper proves hardness results for POHPP under both distance to outerplanar and edge clique cover number, but does not establish para-hardness for either parameter. The authors specifically note that the two complexity statuses are not completely clarified, expressing optimism about an FPT algorithm for distance to outerplanar while describing the edge clique cover case as less clear.

References

For both parameters, POHPP is -hard but it is open whether it is also para--hard.

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