Novel Triple-Based Problems for the Construction of Phylogenetic Networks via Least Common Ancestors

2026-06-23

AI Summary

This study addresses the challenge of inferring rooted phylogenetic networks from incomplete genomic data using limited rooted triplets (e.g., xy|z). It formally introduces, for the first time, the LCA-based triplet consistency problem and proposes the novel concept of “anchored triplets,” thereby overcoming limitations inherent in traditional tree models. By reframing network construction as a constrained LCA realization problem, the authors employ graph-theoretic modeling and directed acyclic graph (DAG) construction to design polynomial-time algorithms that handle both mandatory and forbidden LCA constraints. All variants of the problem are shown to be decidable in polynomial time, and whenever feasible, the approach efficiently outputs biologically meaningful phylogenetic networks consistent with the input triplets.

Abstract

Evolutionary histories are often represented by rooted phylogenetic networks, whose leaves correspond to extant taxa and whose internal vertices represent ancestral lineages. Since such histories must usually be inferred from incomplete data, in particular from genomic sequences of present-day taxa, one often obtains only local information about relative evolutionary proximity. For instance, sequence data may suggest that two taxa $x$ and $y$ are more closely related to each other than either is to a third taxon $z$. This information is classically encoded by a rooted triple $xy|z$. In this paper, we study rooted triples in phylogenetic networks under an ancestor-based interpretation: $xy|z$ is displayed if the unique least common ancestor (LCA) of $x$ and $y$ lies strictly below the unique LCA of $x$ and $z$, respectively of $y$ and $z$, and the latter two LCAs coincide. We also introduce anchored triples $\underline{x}y|z$, which retain only the asymmetric comparison that the LCA of $x$ and $y$ lies below the LCA of $x$ and $z$. This relaxation is natural in networks, where different pairwise ancestral relationships need not behave as they do in trees. We consider several variants of consistency problems for ordinary and anchored triples, both with and without forbidden triples. Somewhat surprisingly, these ancestor-based consistency questions for triples in phylogenetic networks do not appear to have been addressed before despite their direct biological interpretation and the fact that such constraints can be inferred naturally from genomic sequence data. By translating these questions into realization problems for required and forbidden LCA-constraints, we show that all resulting problems can be solved in polynomial time. Moreover, whenever a solution exists, a suitable realizing DAG and phylogenetic network can be constructed within the same time bound.

Problem

• phylogenetic networks
• rooted triples
• least common ancestors
• consistency problems
• anchored triples

Innovation

• phylogenetic networks
• least common ancestor
• rooted triples
• anchored triples
• polynomial-time algorithm