Attaching leaves and picking cherries to characterise the hybridisation number for a set of phylogenies

TL;DR

This paper advances the understanding of hybridisation numbers in phylogenetic networks by providing new characterisations for arbitrary sets of trees, extending previous work and highlighting computational limitations.

Contribution

It introduces a novel characterisation for the minimum hybridisation number for any set of phylogenetic trees, extending to non-binary trees and general network spaces.

Findings

New characterisation for hybridisation number in tree-child networks

Extension of characterisation to all rooted phylogenetic networks

Identification of computational hardness limitations

Abstract

Throughout the last decade, we have seen much progress towards characterising and computing the minimum hybridisation number for a set P of rooted phylogenetic trees. Roughly speaking, this minimum quantifies the number of hybridisation events needed to explain a set of phylogenetic trees by simultaneously embedding them into a phylogenetic network. From a mathematical viewpoint, the notion of agreement forests is the underpinning concept for almost all results that are related to calculating the minimum hybridisation number for when |P|=2. However, despite various attempts, characterising this number in terms of agreement forests for |P|>2 remains elusive. In this paper, we characterise the minimum hybridisation number for when P is of arbitrary size and consists of not necessarily binary trees. Building on our previous work on cherry-picking sequences, we first establish a new characterisation to compute the minimum hybridisation number in the space of tree-child networks. Subsequently, we show how this characterisation extends to the space of all rooted phylogenetic networks. Moreover, we establish a particular hardness result that gives new insight into some of the limitations of agreement forests.