A $\mu$-distance for semidirected orchard phylogenetic networks

TL;DR

This paper introduces a new polynomial-time computable distance measure for comparing orchard binary semidirected phylogenetic networks, enhancing tools for evolutionary biology analysis.

Contribution

It extends the $$-representation to semidirected networks, enabling distinction and comparison of orchard networks with a reconstruction algorithm.

Findings

Introduces a new edge-based $$-representation for semidirected networks.

Provides a polynomial-time reconstruction algorithm for orchard networks.

Establishes a true distance measure for orchard binary semidirected networks.

Abstract

In evolutionary biology, phylogenetic networks are now widely used to represent the historical relationships between species and population, when this history includes reticulation events such as hybridization, gene flow and admixture between populations. Semidirected phylogenetic networks are appropriate models when the direction of some edges and the root position are not identifiable from data. Comparing semidirected networks is important in many applications. For rooted and directed networks, a $\mu$-representation was originally introduced to distinguish tree-child networks, and has since been extended in two different directions: to the larger class of orchard directed networks by adding an extra component that counts paths to reticulations; and to semidirected networks, through an edge-based variant. However, the latter does not provide a distance between semidirected and orchard networks. We introduce here a new edge-based $\mu$-representation capable of distinguishing distinct orchard binary semidirected networks. For this class, we provide a reconstruction algorithm and therefore obtain a true distance that is computable in polynomial time.