Determine the computational complexity and approximability of unresolved parameters

Determine the computational complexity of computing the two graph parameters identified by the paper as unresolved in item 3, and establish whether non-trivial inapproximability results hold for those parameters.

Background

The paper contrasts parameters that are NP-hard to compute exactly with parameters that are computable in polynomial time and provide constant-factor approximations. It then singles out two additional parameters for which the computational status is not settled in the paper.

The problem has two parts: classify the exact computational complexity of the specified parameters and determine whether approximation hardness 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 5, “Concluding remarks and open questions,” item 3