Tightness of the agreement-subgraph bounds for SNPR distance

Determine whether the bounds d_AD(N,N') ≤ d^*_{SNPR}(N,N') ≤ 6d_AD(N,N') are tight for arbitrary rooted binary phylogenetic networks N and N' on the same leaf set, where d^*_{SNPR} is the minimum number of SNPR operations transforming N into N' and d_AD is the minimum number of unlabelled degree-one vertices in a corresponding collection of agreement subgraphs.

Background

Klawitter’s agreement-subgraph framework gives, for any two rooted binary phylogenetic networks, a lower and upper bound on the unweighted SNPR distance: the agreement-subgraph measure d_AD is at most the SNPR distance d^*_{SNPR}, which in turn is at most six times d_AD. The paper explicitly notes that, unlike the bounds established for the weighted tree-child SNPR distance in its main theorem, the tightness of Klawitter’s bounds is unresolved.

Resolving this problem would establish whether either bound can be attained in general, or whether the constant-factor gap in the upper bound can be improved. The question concerns arbitrary rooted binary phylogenetic networks rather than only the tree-child networks studied in the main results of the paper.