Complexity and inapproximability of hard parameters

Determine the computational complexity of the graph parameters identified as NP-hard to compute exactly in the paper and establish whether non-trivial inapproximability results hold for those parameters.

Background

The paper distinguishes parameters that are NP-hard to compute exactly from parameters that are computable in polynomial time and provide constant-factor approximations. It explicitly raises the unresolved problem of determining the precise computational complexity of the hard parameters and asks whether stronger hardness-of-approximation results can be proved.

References

What is the complexity of computing $(G),$ $(G)$}? Can non-trivial inapproximability results be obtained for those hard to compute parameters?

— Graph parameters that are coarsely equivalent to path-length  (2503.05661 - Dragan et al., 7 Mar 2025) in Section 4, “Concluding remarks and open questions,” item 3