Quarnet inference rules for level-1 networks

TL;DR

This paper develops inference rules based on 4-leaved networks, called quarnets, to determine when a collection can be represented by a level-1 phylogenetic network, advancing methods for inferring complex evolutionary histories.

Contribution

It introduces novel inference rules using quarnets to characterize when a set of 4-element subsets can be displayed by a level-1 network, extending previous tree-based methods.

Findings

Rules can characterize collections displayed by level-1 networks.

Quarnets enable inference of cyclic orderings of species.

New directions for algorithms in phylogenetic network inference.

Abstract

An important problem in phylogenetics is the construction of phylogenetic trees. One way to approach this problem, known as the supertree method, involves inferring a phylogenetic tree with leaves consisting of a set $X$ of species from a collection of trees, each having leaf-set some subset of $X$. In the 1980's characterizations, certain inference rules were given for when a collection of 4-leaved trees, one for each 4-element subset of $X$, can all be simultaneously displayed by a single supertree with leaf-set $X$. Recently, it has become of interest to extend such results to phylogenetic networks. These are a generalization of phylogenetic trees which can be used to represent reticulate evolution (where species can come together to form a new species). It has been shown that a certain type of phylogenetic network, called a level-1 network, can essentially be constructed from 4-leaved trees. However, the problem of providing appropriate inference rules for such networks remains unresolved. Here we show that by considering 4-leaved networks, called quarnets, as opposed to 4-leaved trees, it is possible to provide such rules. In particular, we show that these rules can be used to characterize when a collection of quarnets, one for each 4-element subset of $X$, can all be simultaneously displayed by a level-1 network with leaf-set $X$. The rules are an intriguing mixture of tree inference rules, and an inference rule for building up a cyclic ordering of $X$ from orderings on subsets of $X$ of size 4. This opens up several new directions of research for inferring phylogenetic networks from smaller ones, which could yield new algorithms for solving the supernetwork problem in phylogenetics.