Folding and unfolding phylogenetic trees and networks

TL;DR

This paper investigates the properties of folding and unfolding operations between phylogenetic networks and MUL-trees, introduces stable networks, and develops a graph fibration approach to relate tree display in networks and MUL-trees.

Contribution

It characterizes stable networks, relates them to tree-sibling networks, and introduces a phylogenetic graph fibration framework connecting tree display and reconciliation.

Findings

Stable networks are characterized and related to tree-sibling networks.

A phylogenetic graph fibration framework is developed.

Connections between displaying trees in networks and MUL-trees are established.

Abstract

Phylogenetic networks are rooted, labelled directed acyclic graphs which are commonly used to represent reticulate evolution. There is a close relationship between phylogenetic networks and multi-labelled trees (MUL-trees). Indeed, any phylogenetic network $N$ can be 'unfolded' to obtain a MUL-tree $U(N)$ and, conversely, a MUL-tree $T$ can in certain circumstances be 'folded' to obtain a phylogenetic network $F(T)$ that exhibits $T$. In this paper, we study properties of the operations $U$ and $F$ in more detail. In particular, we introduce the class of stable networks, phylogenetic networks $N$ for which $F(U(N))$ is isomorphic to $N$, characterise such networks, and show that that they are related to the well-known class of tree-sibling networks. We also explore how the concept of displaying a tree in a network $N$ can be related to displaying the tree in the MUL-tree $U(N)$. To do this, we develop a phylogenetic analogue of graph fibrations. This allows us to view $U(N)$ as the analogue of the universal cover of a digraph, and to establish a close connection between displaying trees in $U(N)$ and reconciling phylogenetic trees with networks.