Tree-based networks: characterisations, metrics, and support trees

TL;DR

This paper extends the theory of tree-based phylogenetic networks from binary to arbitrary networks, providing characterisations, metrics, and formulas for support trees, enhancing understanding of evolutionary history representations.

Contribution

It generalizes recent characterisations and measures for binary networks to arbitrary phylogenetic networks, and characterizes support trees with explicit formulas.

Findings

Extended characterisations using bipartite matchings, path partitions, and antichains.

Provided explicit formulas for the number of support trees in binary networks.

Generalized concepts to non-binary phylogenetic networks.

Abstract

Phylogenetic networks generalise phylogenetic trees and allow for the accurate representation of the evolutionary history of a set of present-day species whose past includes reticulate events such as hybridisation and lateral gene transfer. One way to obtain such a network is by starting with a (rooted) phylogenetic tree $T$, called a base tree, and adding arcs between arcs of $T$. The class of phylogenetic networks that can be obtained in this way is called tree-based networks and includes the prominent classes of tree-child and reticulation-visible networks. Initially defined for binary phylogenetic networks, tree-based networks naturally extend to arbitrary phylogenetic networks. In this paper, we generalise recent tree-based characterisations and associated proximity measures for binary phylogenetic networks to arbitrary phylogenetic networks. These characterisations are in terms of matchings in bipartite graphs, path partitions, and antichains. Some of the generalisations are straightforward to establish using the original approach, while others require a very different approach. Furthermore, for an arbitrary tree-based network $N$, we characterise the support trees of $N$, that is, the tree-based embeddings of $N$. We use this characterisation to give an explicit formula for the number of support trees of $N$ when $N$ is binary. This formula is written in terms of the components of a bipartite graph.