Finite covering number for orchard networks

Characterize the orchard phylogenetic networks that have finite orchard covering number.

Background

The appendix gives an orchard network and a subtree that cannot be reduced by any cherry-picking sequence of the network, showing that orchard covering numbers need not be finite. The paper therefore leaves unresolved a structural characterization of precisely those orchard networks for which a finite covering set exists.

References

For the former, it would be interesting to characterize orchard networks with finite covering number.

— 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”