On the challenge of reconstructing level-1 phylogenetic networks from triplets and clusters

TL;DR

This paper investigates the difficulty of uniquely reconstructing level-1 phylogenetic networks from triplet and cluster data, revealing limitations and identifying subclasses where reconstruction is possible, with implications for phylogenetic inference algorithms.

Contribution

It demonstrates that unique reconstruction from triplets or clusters is not always possible for level-1 networks and introduces subclasses where it is achievable, along with enumerative results.

Findings

Unique reconstruction is not always possible from all triplets or clusters.

A subclass of level-1 networks can be uniquely reconstructed from their triplet/cluster systems.

Certain subclasses can be reconstructed from proper subsets of triplets and clusters.

Abstract

Phylogenetic networks have gained prominence over the years due to their ability to represent complex non-treelike evolutionary events such as recombination or hybridization. Popular combinatorial objects used to construct them are triplet systems and cluster systems, the motivation being that any network $N$ induces a triplet system $\mathcal R(N)$ and a softwired cluster system $\mathcal S(N)$. Since in real-world studies it cannot be guaranteed that all triplets/softwired clusters induced by a network are available it is of particular interest to understand whether subsets of $\mathcal R(N)$ or $\mathcal S(N)$ allow one to uniquely reconstruct the underlying network $N$. Here we show that even within the highly restricted yet biologically interesting space of level-1 phylogenetic networks it is not always possible to uniquely reconstruct a level-1 network $N$ even when all triplets in $\mathcal R(N)$ or all clusters in $\mathcal S(N)$ are available. On the positive side, we introduce a reasonably large subclass of level-1 networks the members of which are uniquely determined by their induced triplet/softwired cluster systems. Along the way, we also establish various enumerative results, both positive and negative, including results which show that certain special subclasses of level-1 networks $N$ can be uniquely reconstructed from proper subsets of $\mathcal R(N)$ and $\mathcal S(N)$. We anticipate these results to be of use in the design of, for example, algorithms for phylogenetic network inference.