Hypercubes and Hamilton cycles of display sets of rooted phylogenetic networks

TL;DR

This paper investigates the combinatorial structure of phylogenetic trees embedded in networks, establishing conditions for their reconstruction and revealing Hamiltonian cycles in associated graphs, with implications for efficient network inference.

Contribution

It introduces new combinatorial properties of embedded trees, characterizes when they form hypercubes, and provides a polynomial-time algorithm for reconstructing specific phylogenetic networks.

Findings

The rSPR graph of embedded trees has a Hamiltonian cycle when its size is a power of two.

Necessary and sufficient conditions are identified for embedding trees in level-1 networks without extra trees.

A polynomial-time algorithm is developed for reconstructing such networks if conditions are met.

Abstract

In the context of reconstructing phylogenetic networks from a collection of phylogenetic trees, several characterisations and subsequently algorithms have been established to reconstruct a phylogenetic network that collectively embeds all trees in the input in some minimum way. For many instances however, the resulting network also embeds additional phylogenetic trees that are not part of the input. However, little is known about these inferred trees. In this paper, we explore the relationships among all phylogenetic trees that are embedded in a given phylogenetic network. First, we investigate some combinatorial properties of the collection P of all rooted binary phylogenetic trees that are embedded in a rooted binary phylogenetic network N. To this end, we associated a particular graph G, which we call rSPR graph, with the elements in P and show that, if |P|=2^k, where k is the number of vertices with in-degree two in N, then G has a Hamiltonian cycle. Second, by exploiting rSPR graphs and properties of hypercubes, we turn to the well-studied class of rooted binary level-1 networks and give necessary and sufficient conditions for when a set of rooted binary phylogenetic trees can be embedded in a level-1 network without inferring any additional trees. Lastly, we show how these conditions translate into a polynomial-time algorithm to reconstruct such a network if it exists.