Trinets encode tree-child and level-2 phylogenetic networks (Q2249670)

Phylogenetic networks generalize evolutionary trees, and are commonly used to represent evolutionary histories of species that undergo reticulate evolutionary processes such as hybridization, recombination and lateral gene transfer. There has been great interest in trying to development hods to construct rooted phylogenetic networks from triplets, that is, rooted trees on three species. However, although triplets determine or encode rooted phylogenetic trees, they do not in general encode rooted phylogenetic networks, which is a potential issue for any such method. \textit{K. T. Huber} and \textit{V. Moulton} [Algorithmica 66, No. 3, 714--738 (2013; Zbl 1267.05240)] introduced trinets as a natural extension of rooted triplets to networks. They showed that level-1 phylogenetic networks are encoded by their trinets, and also conjectured that all ``recoverable'' rooted phylogenetic networks are encoded by their trinets. In this article, the authors prove that recoverable binary level-2 networks and binary tree-child networks are also encoded by their trinets. To do this they prove two decomposition theorems based on trinets which hold for all recoverable binary rooted phylogenetic networks. This results provide some additional evidence in support of the conjecture that trinets encode all recoverable rooted phylogenetic networks, and could also lead to new approaches to construct phylogenetic networks from trinets.