Bridging the gap between rooted and unrooted phylogenetic networks

TL;DR

This paper establishes a theoretical connection between rooted and unrooted phylogenetic networks, specifically linking 1-nested rooted networks with median networks, and introduces new characterizations and optimality results.

Contribution

It develops a novel link between 1-nested rooted networks and median networks, enhancing understanding of their interplay and properties.

Findings

A method to derive 1-nested networks from median networks.

Characterization of circular split systems via $ ext{I}$-intersection closure.

An analogue of the Splits Equivalence Theorem for 1-nested networks.

Abstract

The need for structures capable of accommodating complex evolutionary signals such as those found in, for example, wheat has fueled research into phylogenetic networks. Such structures generalize the standard phylogenetic tree model by also allowing cycles and have been introduced in rooted and unrooted form. In contrast to phylogenetic trees, however, surprisingly little is known about the interplay between both types thus hampering our ability to make much needed progress for rooted phylogenetic networks by drawing on insights from their much better understood unrooted counterparts. Unrooted phylogenetic networks are underpinned by split systems and by focusing on them we establish a first link between both types. More precisely, we develop a link between 1-nested phylogenetic networks which are examples of rooted phylogenetic networks and the well-studied median networks (aka Buneman graph) which are examples of unrooted phylogenetic networks. In particular, we show that not only can a 1-nested network be obtained from a median network but also that that network is, in a well-defined sense, optimal. Along the way, we characterize circular split systems in terms of the novel $\mathcal I$-intersection closure of a split system and establish the 1-nested analogue of the fundamental "Splits Equivalence Theorem" for phylogenetic trees.