# Orienting undirected phylogenetic networks

## TL;DR

This paper presents polynomial-time algorithms for orienting undirected phylogenetic networks into directed ones, characterizes when this is possible, and discusses the uniqueness and tractability of such orientations.

## Contribution

It introduces algorithms for orienting undirected phylogenetic networks as directed networks, including cases with known or unknown root and reticulation locations, and characterizes when unique orientations exist.

## Key findings

- Polynomial-time algorithm for orientation with known root and reticulation locations.
- Characterization of when an undirected network can be uniquely oriented.
- Fixed-parameter tractable algorithm for orientation based on network level.

## Abstract

This paper studies the relationship between undirected (unrooted) and directed (rooted) phylogenetic networks. We describe a polynomial-time algorithm for deciding whether an undirected nonbinary phylogenetic network, given the locations of the root and reticulation vertices, can be oriented as a directed nonbinary phylogenetic network. Moreover, we characterize when this is possible and show that, in such instances, the resulting directed nonbinary phylogenetic network is unique. In addition, without being given the location of the root and the reticulation vertices, we describe an algorithm for deciding whether an undirected binary phylogenetic network $N$ can be oriented as a directed binary phylogenetic network of a certain class. The algorithm is fixed-parameter tractable (FPT) when the parameter is the level of $N$ and is applicable to classes of directed phylogenetic networks that satisfy certain conditions. As an example, we show that the well-studied class of binary tree-child networks satisfies these conditions.