Behavior of the SubsetConnectivity function

Determine whether the equality c(n)=n-2 holds for all n greater than or equal to 4, or characterize the values of n for which the behavior changes; equivalently, determine whether c(n+1)=c(n) for some n greater than or equal to 4.

Background

For star phylogenetic trees, the paper proves that the covering number is equivalent to the newly introduced SubsetConnectivity problem. Computational verification establishes c(n)=n-2 for 4≤n≤7, while the paper proves only that c(n+1) is either c(n) or c(n)+1. The unresolved issue is whether the observed equality persists for larger n or whether plateaus and other changes occur.

References

We wonder if this equality holds for larger~$n$, or if the behavior changes for certain values of~$n$. Equivalently, we ask whether~$c(n+1) = c(n)$ for some~$n\ge 4$.

— 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”