Determine the complexity and approximability of two unresolved parameters

Determine the computational complexity of the two graph parameters explicitly listed in the open-question item following the paper’s discussion of exact computation, and establish whether nontrivial inapproximability results hold for those parameters.

Background

The paper distinguishes parameters that are NP-hard to compute exactly from parameters for which polynomial-time algorithms and constant-factor approximations are available. It then identifies two further parameters for which the computational status is left unresolved.

The open problem asks both for complexity classifications and for nontrivial inapproximability results. The source text’s notation for the two parameters is rendered incompletely in the supplied manuscript, but the question appears as a separate numbered item in the concluding open-questions section.

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 5, “Concluding remarks and open questions,” item 3