On the existence of infinitely many universal tree-based networks

TL;DR

This paper proves that for any number of leaves greater than one, there are infinitely many universal tree-based networks that can serve as base structures for all rooted binary phylogenetic trees on those leaves.

Contribution

It establishes the existence of infinitely many universal tree-based networks for any number of leaves greater than one, answering a previously open question.

Findings

Existence of infinitely many universal networks for n > 1

Universal networks can serve as base trees for all rooted binary phylogenetic trees

Addresses a question posed by Francis and Steel

Abstract

A tree-based network on a set $X$ of $n$ leaves is said to be universal if any rooted binary phylogenetic tree on $X$ can be its base tree. Francis and Steel showed that there is a universal tree-based network on $X$ in the case of $n=3$, and asked whether such a network exists in general. We settle this problem by proving that there are infinitely many universal tree-based networks for any $n>1$.