A Wild Sheep Chase Through an Orchard

TL;DR

This paper investigates the computational complexity of recognizing undirected orchard networks, a class of phylogenetic networks, proving that the problem is NP-hard and introducing a new characterization to aid future research.

Contribution

It is the first to analyze the complexity of undirected orchard networks and provides a novel characterization that may facilitate further algorithmic developments.

Findings

Deciding if an undirected network is an orchard is NP-hard.

Introduces a new characterization of undirected orchards.

Highlights the gap in studying undirected orchard networks.

Abstract

Orchards are a biologically relevant class of phylogenetic networks as they can describe treelike evolutionary histories augmented with horizontal transfer events. Moreover, the class has attractive mathematical characterizations that can be exploited algorithmically. On the other hand, undirected orchard networks have hardly been studied yet. Here, we prove that deciding whether an undirected, binary phylogenetic network is an orchard -- or equivalently, whether it has an orientation that makes it a rooted orchard -- is NP-hard. For this, we introduce a new characterization of undirected orchards which could be useful for proving positive results.