Inferring Phylogenetic Networks from Required and Forbidden LCA-Constraints

Phylogenetic networks provide a framework for representing evolutionary histories involving reticulate events such as hybridization or horizontal gene transfer. A central problem is to infer such networks from local structural information. In this paper, we study network inference from least common ancestor (LCA) constraints, which specify relative ancestral relationships between pairs of taxa. While previous work has characterized when a set of required LCA constraints can be realized by a phylogenetic network, practical applications may also involve constraints that must be explicitly avoided, for example due to biological prior knowledge. We therefore consider the realization problem for pairs $(R,F)$, where $R$ is a set of required LCA-constraints and $F$ is a set of forbidden ones. Since there are several natural ways to formalize what it means for a network to avoid a forbidden LCA-constraint, we study three such variants. For each of them, we characterize exactly when there exists a phylogenetic network that realizes all constraints in $R$ while avoiding all constraints in $F$ in the respective sense. Based on these characterizations, we derive polynomial-time algorithms that decide the existence of such networks and construct one whenever it exists.