Trinets encode orchard phylogenetic networks

TL;DR

This paper demonstrates that trinets uniquely encode rooted binary orchard networks and provides a polynomial-time algorithm for reconstructing such networks from their trinets.

Contribution

It establishes that trinets are sufficient to encode rooted binary orchard networks and introduces a polynomial-time reconstruction algorithm.

Findings

Trinets encode rooted binary orchard networks.

A polynomial-time algorithm reconstructs networks from trinets.

Answers affirmatively to polynomial reconstruction from trinets.

Abstract

Rooted triples, rooted binary phylogenetic trees on three leaves, are sufficient to encode rooted binary phylogenetic trees. That is, if $\mathcal T$ and $\mathcal T'$ are rooted binary phylogenetic $X$-trees that infers the same set of rooted triples, then $\mathcal T$ and $\mathcal T'$ are isomorphic. However, in general, this sufficiency does not extend to rooted binary phylogenetic networks. In this paper, we show that trinets, phylogenetic network analogues of rooted triples, are sufficient to encode rooted binary orchard networks. Rooted binary orchard networks naturally generalise rooted binary tree-child networks. Moreover, we present a polynomial-time algorithm for building a rooted binary orchard network from its set of trinets. As a consequence, this algorithm affirmatively answers a previously-posed question of whether there is a polynomial-time algorithm for building a rooted binary tree-child network from the set of trinets it infers.