Non-essential arcs in phylogenetic networks

Authors

Simone Linz, Charles Semple

Categories

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 $Π_2^P$-complete.