Editing orthology graphs into level-1 explainable graphs

Determine the computational complexity of finding the minimum number of edge additions or deletions required to transform an orthology graph that is not level-1 explainable into a level-1 explainable graph.

Background

The paper characterizes level-1 explainable orthology graphs as those whose primitive induced subgraphs are near-cographs. It therefore identifies graph editing as a natural next problem: when an orthology graph fails this structural condition, one may add or remove edges to obtain a level-1 explainable graph. The authors explicitly leave the computational complexity of this editing task unresolved.

References

Clearly, if an orthology graph $G$ lacks this structure, it would be necessary to determine the appropriate edits (removal or addition of edges) required to transform it into a \levIex graph. The computational complexity of the underlying editing problem remain an interesting open question for future research.

— Orthology and Near-Cographs in the Context of Phylogenetic Networks  (2502.08746 - Lindeberg et al., 12 Feb 2025) in Section 8, “Summary and outlook” (Section \ref{sec:outro})