A Note on Encodings of Phylogenetic Networks of Bounded Level

TL;DR

This paper characterizes when collections of phylogenetic trees, clusters, or triplets uniquely encode level-1 phylogenetic networks, providing a complete answer for this specific case and discussing potential extensions.

Contribution

It offers a complete characterization of encodings for level-1 phylogenetic networks, a foundational step in understanding more complex network hierarchies.

Findings

Identifies conditions under which collections encode level-1 networks

Provides a complete characterization for level-1 networks

Shows that the approach does not extend straightforwardly to higher levels

Abstract

Driven by the need for better models that allow one to shed light into the question how life's diversity has evolved, phylogenetic networks have now joined phylogenetic trees in the center of phylogenetics research. Like phylogenetic trees, such networks canonically induce collections of phylogenetic trees, clusters, and triplets, respectively. Thus it is not surprising that many network approaches aim to reconstruct a phylogenetic network from such collections. Related to the well-studied perfect phylogeny problem, the following question is of fundamental importance in this context: When does one of the above collections encode (i.e. uniquely describe) the network that induces it? In this note, we present a complete answer to this question for the special case of a level-1 (phylogenetic) network by characterizing those level-1 networks for which an encoding in terms of one (or equivalently all) of the above collections exists. Given that this type of network forms the first layer of the rich hierarchy of level-k networks, k a non-negative integer, it is natural to wonder whether our arguments could be extended to members of that hierarchy for higher values for k. By giving examples, we show that this is not the case.