Non-essential arcs in phylogenetic networks

TL;DR

This paper introduces the concept of non-essential arcs in phylogenetic networks, providing a characterization for tree-child networks and analyzing the computational complexity of identifying such arcs.

Contribution

It defines non-essential arcs, characterizes them for tree-child networks, and analyzes the complexity of recognizing these arcs in general networks.

Findings

Identifies non-essential arcs in tree-child networks using directed graphs

Shows that deleting all non-essential arcs can be done in cubic time

Proves that deciding if an arc is non-essential is a2_2^P-complete

Abstract

In the study of rooted phylogenetic networks, analyzing the set of rooted phylogenetic trees that are embedded in such a network is a recurring task. From an algorithmic viewpoint, this analysis almost always requires an exhaustive search of a particular multiset $S$ of rooted phylogenetic trees that are embedded in a rooted phylogenetic network $\mathcal{N}$. Since the size of $S$ is exponential in the number of reticulations of $\mathcal{N}$, it is consequently of interest to keep this number as small as possible but without loosing any element of $S$. In this paper, we take a first step towards this goal by introducing the notion of a non-essential arc of $\mathcal{N}$, which is an arc whose deletion from $\mathcal{N}$ results in a rooted phylogenetic network $\mathcal{N}'$ such that the sets of rooted phylogenetic trees that are embedded in $\mathcal{N}$ and $\mathcal{N}'$ are the same. We investigate the popular class of tree-child networks and characterize which arcs are non-essential. This characterization is based on a family of directed graphs. Using this novel characterization, we show that identifying and deleting all non-essential arcs in a tree-child network takes time that is cubic in the number of leaves of the network. Moreover, we show that deciding if a given arc of an arbitrary phylogenetic network is non-essential is $\Pi_2^P$-complete.