Minimum-hybrid realization

Determine the computational complexity of finding a directed acyclic graph or phylogenetic network that realizes a relation R while minimizing the number of hybrid vertices globally or within each biconnected component.

Background

The paper notes that vertex count is not the only possible measure of simplicity for DAGs and phylogenetic networks. Since hybrid vertices represent deviations from tree structure, minimizing their number could provide an alternative optimization objective. The authors explicitly pose this computational problem, including both a global version and a biconnected-component version.

References

We therefore also pose the problem of determining the computational complexity of finding a DAG or network that realizes $R$ with as few hybrids as possible, either globally or per biconnected component.

— Inferring DAGs and Phylogenetic Networks from Least Common Ancestors  (2511.07965 - Lindeberg et al., 11 Nov 2025) in Section 8, Discussion and Outlook, paragraph “Underlying Optimization Problems”