Polynomial-time computation for finite network covering numbers

Determine whether the covering number can be computed in polynomial time whenever the covering number for the network input class is finite.

Background

The paper raises extensions of CoveringNumber from phylogenetic trees to network classes such as orchard and tree-child networks. Since covering numbers for network inputs may be infinite, the unresolved algorithmic question concerns the cases known to have finite covering number: whether the minimum covering set can then be computed efficiently.

References

Moreover, whenever the covering number for the network inputs is known to be finite, can we still compute it in polynomial time?

— How many cherry-picking sequences are needed to reduce all subtrees of a phylogenetic tree?  (2608.19916 - Kollmann et al., 20 Aug 2026) in Section 6, Discussion, paragraph “Open problems”