CSD Homomorphisms Between Phylogenetic Networks

TL;DR

This paper introduces connected surjective digraph maps (CSD) as a way to relate complex phylogenetic networks, providing a framework to analyze and approximate real-world evolutionary relationships beyond simple trees.

Contribution

It defines CSD maps, proves their properties, and explores their implications for understanding and classifying phylogenetic networks.

Findings

CSD maps behave well under composition.

Existence of CSD maps constrains network structure.

CSD maps enable lifting undirected networks into more resolved forms.

Abstract

Since Darwin, species trees have been used as a simplified description of the relationships which summarize the complicated network $N$ of reality. Recent evidence of hybridization and lateral gene transfer, however, suggest that there are situations where trees are inadequate. Consequently it is important to determine properties that characterize networks closely related to $N$ and possibly more complicated than trees but lacking the full complexity of $N$. A connected surjective digraph map (CSD) is a map $f$ from one network $N$ to another network $M$ such that every arc is either collapsed to a single vertex or is taken to an arc, such that $f$ is surjective, and such that the inverse image of a vertex is always connected. CSD maps are shown to behave well under composition. It is proved that if there is a CSD map from $N$ to $M$, then there is a way to lift an undirected version of $M$ into $N$, often with added resolution. A CSD map from $N$ to $M$ puts strong constraints on $N$. In general, it may be useful to study classes of networks such that, for any $N$, there exists a CSD map from $N$ to some standard member of that class.