Is this network proper forest-based?

TL;DR

This paper investigates the computational complexity of determining whether a given network in evolutionary biology can be constructed from a forest of phylogenetic trees, revealing polynomial-time solutions for some cases and NP-completeness for others.

Contribution

It introduces a new characterization based on vertex colorings and establishes complexity results for proper forest-based networks, including a fixed parameter tractable algorithm.

Findings

Polynomial-time decision for binary, tree-child networks with two roots.

NP-completeness for networks with three or more roots.

FPT algorithm for networks with vertices of indegree at most 2.

Abstract

In evolutionary biology, networks are becoming increasingly used to represent evolutionary histories for species that have undergone non-treelike or reticulate evolution. Such networks are essentially directed acyclic graphs with a leaf set that corresponds to a collection of species, and in which non-leaf vertices with indegree 1 correspond to speciation events and vertices with indegree greater than 1 correspond to reticulate events such as gene transfer. Recently forest-based networks have been introduced, which are essentially (multi-rooted) networks that can be formed by adding some arcs to a collection of phylogenetic trees (or phylogenetic forest), where each arc is added in such a way that its ends always lie in two different trees in the forest. In this paper, we consider the complexity of deciding whether or not a given network is proper forest-based, that is, whether it can be formed by adding arcs to some underlying phylogenetic forest which contains the same number of trees as there are roots in the network. More specifically, we show that it can be decided in polynomial time whether or not a binary, tree-child network with $m \ge 2$ roots is proper forest-based in case $m=2$, but that this problem is NP-complete for $m\ge 3$. We also give a fixed parameter tractable (FPT) algorithm for deciding whether or not a network in which every vertex has indegree at most 2 is proper forest-based. A key element in proving our results is a new characterization for when a network with $m$ roots is proper forest-based which is given in terms of the existence of certain $m$-colorings of the vertices of the network.