# Counting Spinal Phylogenetic Networks

## TL;DR

This paper introduces a new method to count specific types of phylogenetic networks using combinatorial structures called expanding covers.

## Contribution

The paper introduces a counting method for spinal phylogenetic networks using expanding covers, a novel combinatorial approach.

## Key findings

- Expanding covers can encode a large class of phylogenetic networks called labellable networks.
- Spinal networks can be fully described using covers and serve as a base case for counting more general network classes.

## Abstract

Phylogenetic networks are an important way to represent evolutionary histories that involve reticulate processes such as hybridisation or horizontal gene transfer, yet fundamental questions such as how many networks there are that satisfy certain properties are very difficult. A new way to encode a large class of networks, using “expanding covers”, may provide a way to approach such problems. Expanding covers encode a large class of phylogenetic networks, called labellable networks. This class does not include all networks, but does include many familiar classes, including orchard, normal, tree-child and tree-sibling networks. As expanding covers are a combinatorial structure, it is possible that they can be used as a tool for counting such classes for a fixed number of leaves and reticulations, for which, in many cases, a closed formula has not yet been found. More recently, a new class of networks was introduced, called spinal networks, which are analogous to caterpillar trees for phylogenetic trees and can be fully described using covers. In the present article, we describe a method for counting networks that are both spinal and belong to some more familiar class, with the hope that these form a base case from which to attack the more general classes.