Counting spinal phylogenetic networks

TL;DR

This paper addresses the problem of counting spinal phylogenetic networks, a tractable subclass of labellable networks encoded via expanding covers. By exploiting a spine-based encoding and bijections to combinatorial structures, it derives explicit counts and recurrences for several binary spinal network classes, including tree-child, stack-free, and fully tree-sibling variants, as well as broader families allowing or forbidding parallel edges. Central results provide closed forms such as $S_{n,| C|} = 2^{\binom{| C|-2}{2}}(|\nC|^n - (|\nC|-1)^n)$ for all spinal networks, and specialized formulas involving Stirling numbers of both kinds and Bessel polynomials for the binary subclasses, e.g., $|\mathcal{STC}_{n,k}| = n!|B_{n-1,k}| - \frac{n!}{2}|B_{n-2,k}|$ with $|B_{n-1,k}| = \frac{(n-1+k)!}{2^k (n-1-k)! k!}$. The work lays a robust counting framework that can serve as a base for attacking more general classes (like tree-child networks) and suggests future recursive approaches related to classical conjectures (e.g., Pons-Batle style recurrences). Overall, it demonstrates the power of the expanding-cover representation to yield concrete, scalable enumeration for complex phylogenetic-network classes.

Abstract

Phylogenetic networks are an important way to represent evolutionary histories that involve reticulations such as hybridization 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.

Figures (4)

• Figure 1: An example of a binary spinal tree-child network with interior vertices labelled according to the tree-child labelling scheme described in Theorem \ref{['t:binspinTCN']}. Each vertex is drawn according to which category they are in. Category $1$ are plain circles, Category $2$ are bold, and Category $3$ are dashed.
• Figure 2: An example of a binary spinal stack-free network with interior vertices labelled according to the stack-free labelling scheme described in the proof of Theorem \ref{['t:spinalsfn']}. Each vertex is drawn according to which category they are in. Category $1$ are plain circles, Category $2$ are dashed, Category $3$ are bold, and Category $4$ are dotted. This example is mapped to the partition $124|3$.
• Figure 3: All binary spinal network shapes with $n=2$ leaves and $k=2$ reticulations. Networks (a) and (b) are fully tree-sibling but not stack-free, while (c) is stack-free but not fully tree-sibling. Networks (d), (e), and (f) are neither fully tree-sibling or stack-free. Note, there are no tree-child networks with $n=k=2$. There are two leaf-labelled copies of each of shapes (a) to (e), while (f) has only one.
• Figure 4: An example of a binary spinal fully tree-sibling network with interior vertices labelled according to the tree-sibling labelling scheme. Each vertex is drawn according to which category they are in. Category $1$ are plain circles, Category $2$ are dashed, and Category $3$ are bold. This example is mapped to, in cycle notation, $(143)(2)$.

Theorems & Definitions (32)

• Definition 2.1
• Definition 2.2: francis2023labellable
• Theorem 2.3: francis2023labellable, Theorem 4.4
• Theorem 2.4: francis2023phylogenetic, Theorem 8.1
• Lemma 3.1
• proof
• Proposition 3.2
• proof
• Definition 3.3: cardona2008comparison
• Theorem 3.4: francis2023phylogenetic, Theorem 4.1