Reconstructing Tree-Child Networks from Reticulate-Edge-Deleted Subnetworks

TL;DR

This paper demonstrates that level-$k$ tree-child networks can be uniquely reconstructed from their reticulate-edge-deleted subnetworks and provides a polynomial-time algorithm for doing so, advancing phylogenetic network reconstruction methods.

Contribution

It introduces a method to reconstruct level-$k$ tree-child networks from specific subnetworks and offers a polynomial-time algorithm for this process.

Findings

Networks are encoded by their reticulate-edge-deleted subnetworks for $k extgreater=2$.

A polynomial-time reconstruction algorithm is provided.

Reconstruction is possible even with subnetworks from each biconnected component.

Abstract

Network reconstruction lies at the heart of phylogenetic research. Two well studied classes of phylogenetic networks include tree-child networks and level-$k$ networks. In a tree-child network, every non-leaf node has a child that is a tree node or a leaf. In a level-$k$ network, the maximum number of reticulations contained in a biconnected component is $k$. Here, we show that level-$k$ tree-child networks are encoded by their reticulate-edge-deleted subnetworks, which are subnetworks obtained by deleting a single reticulation edge, if $k\geq 2$. Following this, we provide a polynomial-time algorithm for uniquely reconstructing such networks from their reticulate-edge-deleted subnetworks. Moreover, we show that this can even be done when considering subnetworks obtained by deleting one reticulation edge from each biconnected component with $k$ reticulations.