# Display sets of normal and tree-child networks

## TL;DR

This paper presents a polynomial-time algorithm for deciding if a normal network and a tree-child network embed the same set of phylogenetic trees, contrasting with the general problem's computational hardness.

## Contribution

The paper introduces a quadratic-time algorithm for a specific case of the phylogenetic network embedding problem involving normal and tree-child networks.

## Key findings

- Polynomial-time algorithm for normal and tree-child networks
- Quadratic running time in leaf set size
- Contrasts with general problem's $ ext{Pi}^P_2$-completeness

## Abstract

Phylogenetic trees canonically arise as embeddings of phylogenetic networks. We recently showed that the problem of deciding if two phylogenetic networks embed the same sets of phylogenetic trees is computationally hard, \blue{in particular, we showed it to be $\Pi^P_2$-complete}. In this paper, we establish a polynomial-time algorithm for this decision problem if the initial two networks consists of a normal network and a tree-child network. The running time of the algorithm is quadratic in the size of the leaf sets.