# A class of phylogenetic networks reconstructable from ancestral profiles

## TL;DR

This paper introduces orchard networks, a class of phylogenetic networks, and provides a polynomial-time method to uniquely reconstruct them from ancestral profiles, extending previous results to more complex networks with reticulation.

## Contribution

The paper defines orchard networks and proves they can be uniquely reconstructed from ancestral profiles using a polynomial-time algorithm, generalizing prior reconstruction results.

## Key findings

- Reconstruction is possible for orchard networks from ancestral profiles.
- A polynomial-time algorithm is provided for reconstructing orchard networks.
- The class of orchard networks includes several previously studied network types.

## Abstract

Rooted phylogenetic networks provide an explicit representation of the evolutionary history of a set $X$ of sampled species. In contrast to phylogenetic trees which show only speciation events, networks can also accommodate reticulate processes (for example, hybrid evolution, endosymbiosis, and lateral gene transfer). A major goal in systematic biology is to infer evolutionary relationships, and while phylogenetic trees can be uniquely determined from various simple combinatorial data on $X$, for networks the reconstruction question is much more subtle. Here we ask when can a network be uniquely reconstructed from its `ancestral profile' (the number of paths from each ancestral vertex to each element in $X$). We show that reconstruction holds (even within the class of all networks) for a class of networks we call `orchard networks', and we provide a polynomial-time algorithm for reconstructing any orchard network from its ancestral profile. Our approach relies on establishing a structural theorem for orchard networks, which also provides for a fast (polynomial-time) algorithm to test if any given network is of orchard type. Since the class of orchard networks includes tree-sibling tree-consistent networks and tree-child networks, our result generalise reconstruction results from 2008 and 2009. Orchard networks allow for an unbounded number $k$ of reticulation vertices, in contrast to tree-sibling tree-consistent networks and tree-child networks for which $k$ is at most $2|X|-4$ and $|X|-1$, respectively.