Which phylogenetic networks are merely trees with additional arcs?

TL;DR

This paper provides a polynomial-time method to determine if a binary phylogenetic network can be derived from a tree by adding arcs, and offers algorithms to find such trees when they exist, advancing understanding of network-tree relationships.

Contribution

It introduces a precise, testable criterion based on 2-SAT for identifying networks that are trees with added arcs, along with an efficient algorithm to find corresponding trees.

Findings

A 2-SAT based criterion for network-tree realizability

Polynomial-time algorithm for finding underlying trees

Several corollaries and open questions are presented

Abstract

A binary phylogenetic network may or may not be obtainable from a tree by the addition of directed edges (arcs) between tree arcs. Here, we establish a precise and easily tested criterion (based on `2-SAT') that efficiently determines whether or not any given network can be realized in this way. Moreover, the proof provides a polynomial-time algorithm for finding one or more trees (when they exist) on which the network can be based. A number of interesting consequences are presented as corollaries; these lead to some further relevant questions and observations, which we outline in the conclusion.