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Counting Phylogenetic Networks with Few Reticulation Vertices: Galled and Reticulation-Visible Networks.

Yu-Sheng Chang¹, Michael Fuchs²

¹Department of Mathematical Sciences, National Chengchi University, Taipei, 116, Taiwan.

Bulletin of Mathematical Biology

|May 18, 2024

Summary

This summary is machine-generated.

We provide exact counts for galled networks and reticulation-visible networks with few reticulation vertices using novel graph and generating function methods. These findings advance the mathematical understanding of complex network structures.

Keywords:

Component graph method Enumeration Galled network Phylogenetic network Reticulation-visible network

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Area of Science:

Combinatorics
Graph Theory
Computational Biology

Background:

Understanding the structure of biological networks is crucial.
Galled networks and reticulation-visible networks are key models in phylogenetics.
Counting these network types with limited reticulation is computationally challenging.

Purpose of the Study:

To derive exact and asymptotic counting results for galled networks and reticulation-visible networks.
To analyze networks with a small number of reticulation vertices.
To explore maximally reticulated reticulation-visible networks.

Main Methods:

Component graph method.
Generating function techniques.
Analytic combinatorics for galled networks.

Main Results:

Exact and asymptotic formulas for counting galled networks with few reticulation vertices.
Exact and asymptotic formulas for counting reticulation-visible networks with few reticulation vertices.
Formulas for maximally reticulated reticulation-visible networks.

Conclusions:

The component graph method and generating functions provide powerful tools for network enumeration.
The study offers precise mathematical insights into the complexity of biological network models.
Results contribute to the theoretical foundation for analyzing evolutionary networks.