Rooted binary level-3 phylogenetic networks are encoded by quarnets

Abstract

Phylogenetic networks generalize evolutionary trees and are commonly used to represent evolutionary relationships between species that undergo reticulate processes such as hybridization, recombination and lateral gene transfer. Recently, there has been great interest in knowing which networks are determined or encoded by their trinets, that are rooted networks on three species. Van Iersel and Moulton showed that recoverable rooted binary level-2 phylogenetic networks are encoded by their trinets. Based on their work for level-2 networks, we show here that not all recoverable rooted binary level-3 networks are weakly encoded by their trinets, but most networks are. Further, although not all level-3 networks are weakly encoded by their trinets, we are able to prove that all recoverable rooted binary level-3 networks are encoded by their quarnets, that are rooted networks on four species.