Restricted trees: simplifying networks with bottlenecks

TL;DR

This paper introduces a method to simplify complex phylogenetic networks into trees by identifying bottlenecks and using restricted connected surjective digraph maps, aiding in understanding fundamental evolutionary relationships.

Contribution

It presents a uniform procedure to construct a 'restricted tree' from a network using restricted CSD maps, capturing key relationships and simplifying network analysis.

Findings

The restricted tree preserves many relationships from the original network.

The procedure provides a well-defined, simplified representation of complex networks.

Many relationships in the tree also appear in the original network.

Abstract

Suppose N is a phylogenetic network indicating a complicated relationship among individuals and taxa. Often of interest is a much simpler network, for example, a species tree T, that summarizes the most fundamental relationships. The meaning of a species tree is made more complicated by the recent discovery of the importance of hybridizations and lateral gene transfers. Hence it is desirable to describe uniform well-defined procedures that yield a tree given a network N. A useful tool toward this end is a connected surjective digraph (CSD) map f from N to N' where N' is generally a much simpler network than N. A set W of vertices in N is "restricted" if there is at most one vertex from which there is an arc into W, thus yielding a bottleneck in N. A CSD map f from N to N' is "restricted" if the inverse image of each vertex in N' is restricted in N. This paper describes a uniform procedure that, given a network N, yields a well-defined tree called the "restricted tree" of N. There is a restricted CSD map from N to the restricted tree. Many relationships in the tree can be proved to appear also in N.