Computational complexity of minimum level explainability

Determine whether finding the smallest integer k such that a given graph G can be explained by a 0/1-labeled level-k network is an NP-hard problem.

Background

The paper establishes that every graph can be explained by a 0/1-labeled phylogenetic level-k network for some k<|X|, and gives a construction based on a vertex set Y whose deletion leaves a cograph. Finding a minimum such deletion set is already known to be NP-hard, but the authors emphasize that the minimum size of such a set need not equal the minimum network level k. Consequently, the computational complexity of directly determining the smallest explainability level remains unresolved.

References

It remains, however, an open question whether the problem of finding the smallest $k$ such that $G$ is \levIex[$k$] is an NP-hard task as well.

— Orthology and Near-Cographs in the Context of Phylogenetic Networks  (2502.08746 - Lindeberg et al., 12 Feb 2025) in Section 3, subsection “Novel constructions,” immediately following Theorem 3.5 (Theorem \ref{thm:every_G_lev-k-ex})