Edge-deletion proximity for tree-basedness

Determine the minimum number of selected edges whose subsequent deletion transforms a rooted phylogenetic network into a tree-based network, even when the selected edges are required not to be reticulation edges at the time of removal.

Background

The paper reviews six proximity measures for rooted phylogenetic networks. The sixth asks for the minimum number of edges, together with subsequent edge subdivisions, that must be deleted to make a network tree-based. The authors state that this optimization problem remains unresolved even under the restriction that only selected edges whose removal produces a tree-based network and that are not reticulation edges at the time of removal may be used. They also note that the optimal values of this edge-deletion measure and the leaf-addition measure are incomparable.

References

Lastly, \cref{rootedProx}.6 is an open problem even if constrained to selected edges whose subsequent removal transforms $N$ into a tree-based network and who, at the time of removal, are not reticulation edges.

— The nearest neighbor interchange proximity measure for the tree-basedness of rooted binary phylogenetic networks  (2610.23426 - Frohn et al., 16 Oct 2026) in Section 2.2, subsection “A review of known tree-basedness proximity measures on rooted phylogenetic networks” (discussion following Problem 1)