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.

翻译：系统发育网络为表示涉及杂交或水平基因转移等网状事件的进化历史提供了框架。一个核心问题是从局部结构信息推断此类网络。本文研究了基于最近共同祖先（LCA）约束的网络推断，这些约束规定了分类群对之间的相对祖先关系。此前研究已刻画了当一组必需的LCA约束可由系统发育网络实现时的条件，但实际应用可能也涉及必须明确避免的约束，例如基于生物学先验知识。因此，我们考虑成对集合$(R,F)$的实现问题，其中$R$是必需的LCA约束集，$F$是被禁止的LCA约束集。由于存在多种自然方式形式化网络避免被禁止的LCA约束的意义，我们研究了三种变体。对于每种变体，我们精确刻画了何时存在一个系统发育网络，该网络在相应意义上实现$R$中的所有约束同时避免$F$中的所有约束。基于这些刻画，我们推导出多项式时间算法，用于判断此类网络的存在性，并在存在时构建一个网络。