Planar Rooted Phylogenetic Networks

TL;DR

This paper studies the properties of planar rooted phylogenetic networks, introduces subclasses, and explores their planarity characteristics, linking phylogenetics with planar graph theory to aid visualization and analysis.

Contribution

It defines a hierarchy of planar rooted phylogenetic networks, characterizes planarity in level-k networks, and establishes a link between regular networks and pyramidal structures.

Findings

Level-1, -2, -3 networks are always outer, terminal, and upward planar.

Level-4 networks are not necessarily planar.

A regular network is terminal planar if and only if it is pyramidal.

Abstract

A rooted phylogenetic network is a directed acyclic graph with a single root, whose sinks correspond to a set of species. As such networks are useful for representing the evolution of species that have undergone reticulate evolution, there has been great interest in developing the theory behind and algorithms for constructing them. However, unlike evolutionary trees, these networks can be highly non-planar, which can make them difficult to visualise and interpret. Here we investigate properties of planar rooted phylogenetic networks and algorithms for deciding whether or not rooted networks have certain special planarity properties. In particular, we introduce three natural subclasses of planar rooted phylogenetic networks and show that they form a hierarchy. In addition, for the well-known level-k networks, we show that level-1, -2, -3 networks are always outer, terminal, and upward planar, respectively, and that level-4 networks are not necessarily planar. Finally, we show that a regular network is terminal planar if and only if it is pyramidal. Our results make use of the highly developed field of planar digraphs, and we believe that the link between phylogenetic networks and planar graphs should prove useful in future for developing new approaches to both construct and visualise phylogenetic networks.