Exploring spaces of semi-directed phylogenetic networks

TL;DR

This paper introduces a new rearrangement move called cut edge transfer (CET) for semi-directed phylogenetic networks, proving the connectivity of their search space, which is crucial for network reconstruction algorithms.

Contribution

It proposes CET and related moves, establishing the connectedness of semi-directed phylogenetic network spaces, enabling comprehensive search and reconstruction.

Findings

CET connects all semi-directed level-1 networks with fixed reticulations.

CET$^+$ and CET$^-$ connect all semi-directed networks with varying reticulations.

Rooted level-1 networks are also connected under CET.

Abstract

Semi-directed phylogenetic networks have recently emerged as a class of phylogenetic networks sitting between rooted (directed) and unrooted (undirected) phylogenetic networks as they contain both directed as well as undirected edges. While the spaces of rooted phylogenetic networks and unrooted phylogenetic networks have been analyzed in recent years and various rearrangement moves to traverse these spaces have been introduced, the results do not immediately carry over to semi-directed phylogenetic networks. Here, we propose a simple rearrangement move for semi-directed phylogenetic networks, called cut edge transfer (CET), and show that the space of semi-directed level-$1$ networks with precisely $k$ reticulations is connected under CET. This level-$1$ space is currently the predominantly used search space for most algorithms that reconstruct semi-directed phylogenetic networks. Hence, every semi-directed level-$1$ network with a fixed number of reticulations and leaf set can be reached from any other such network by a sequence of CETs. By introducing two additional moves, CET$^+$ and CET$^-$, that allow for the addition or deletion of reticulations, we then establish connectedness for the space of all semi-directed phylogenetic networks on a fixed leaf set. As a byproduct of our results for semi-directed phylogenetic networks, we also show that the space of rooted level-$1$ networks with a fixed number of reticulations and leaf set is connected under CET, when translated into the rooted setting.