Merging Arcs to Produce Acyclic Phylogenetic Networks and Normal Networks

Stephen J Willson

doi:10.1007/s11538-021-00986-1

Abstract

As phylogenetic networks grow increasingly complicated, systematic methods for simplifying them to reveal properties will become more useful. This paper considers how to modify acyclic phylogenetic networks into other acyclic networks by contracting specific arcs that include a set D. The networks need not be binary, so vertices in the networks may have more than two parents and/or more than two children. In general, in order to make the resulting network acyclic, additional arcs not in D must also be contracted. This paper shows how to choose D so that the resulting acyclic network is “pre-normal”. As a result, removal of all redundant arcs yields a normal network. The set D can be selected based only on the geometry of the network, giving a well-defined normal phylogenetic network depending only on the given network. There are CSD maps relating most of the networks. The resulting network can be visualized as a “wired lift” in the original network, which appears as the original network with each arc drawn in one of three ways.

Highlights

A phylogenetic tree is a directed tree whose vertices represent biological species, whose leaves typically correspond to known extant species, and whose branchings indicate speciation events, usually by genetic mutation
In the last decades it has become clear that other events such as hybridization and lateral gene transfer are important in evolution, even though they are not modeled using phylogenetic trees
Overviews of phylogenetic networks may be found in Steel (2016) and Huson et al (2010)

Summary

Introduction

A phylogenetic tree is a directed tree whose vertices represent biological species, whose leaves typically correspond to known extant species, and whose branchings indicate speciation events, usually by genetic mutation. For hand calculation the following is often easier: Given N and D, since K (D) must be closed by Theorem 3.6, we adjoin to D all arcs in any directed path between two vertices u and v such that u ∼D v. By Theorem 2.1 a path from v to φ(x) in N of maximal length contains no redundant arc, lies in R(N ) It follows that x ∈ cl(v; R(N )). Theorem 6.2 Let N = (V , A, ρ, φ) and N = (V , A , ρ , φ ) be X -networks, with f : V → V a connected map, and suppose ( f −1, E1) is a wired lift of f.

Comopyrum

Discussion

Conflict of interest None