When are quarnets sufficient to reconstruct semi-directed phylogenetic networks?

TL;DR

This paper investigates when semi-directed phylogenetic networks can be reconstructed from smaller 4-leaf subnetworks called quarnets, establishing that level-2 networks are fully determined by quarnets, unlike level-3.

Contribution

It proves that semi-directed binary level-2 phylogenetic networks are uniquely determined by their quarnets, and that the blob tree is always encoded by quarnets, advancing network reconstruction methods.

Findings

Level-2 networks are encoded by quarnets.

Level-3 networks are not necessarily encoded by quarnets.

Blob tree structure is always encoded by quarnets.

Abstract

Phylogenetic networks are graphs that are used to represent evolutionary relationships between different taxa. They generalize phylogenetic trees since for example, unlike trees, they permit lineages to combine. Recently, there has been rising interest in semi-directed phylogenetic networks, which are mixed graphs in which certain lineage combination events are represented by directed edges coming together, whereas the remaining edges are left undirected. One reason to consider such networks is that it can be difficult to root a network using real data. In this paper, we consider the problem of when a semi-directed phylogenetic network is defined or encoded by the smaller networks that it induces on the 4-leaf subsets of its leaf set. These smaller networks are called quarnets. We prove that semi-directed binary level-2 phylogenetic networks are encoded by their quarnets, but that this is not the case for level-3. In addition, we prove that the so-called blob tree of a semi-directed binary network, a tree that give the coarse-grained structure of the network, is always encoded by the quarnets of the network. These results are relevant for proving the statistical consistency of programs that are currently being developed for reconstructing phylogenetic networks from practical data, such as the recently developed Squirrel software tool.