# Displaying trees across two phylogenetic networks

## TL;DR

This paper investigates the computational complexity of comparing the display sets of two phylogenetic networks, revealing NP-completeness and $ ext{P}^{ ext{NP}}_{||}$-completeness results for key problems in phylogenetics.

## Contribution

It establishes the hardness of determining common trees and equality of display sets for two phylogenetic networks, including the first proof of $ ext{P}^{ ext{NP}}_{||}$-completeness for these problems.

## Key findings

- Deciding if two networks share a common displayed tree is NP-complete.
- Checking if two networks have identical display sets is $ ext{P}^{ ext{NP}}_{||}$-complete in general.
- Some special cases allow polynomial-time solutions, but the general problems are computationally hard.

## Abstract

Phylogenetic networks are a generalization of phylogenetic trees to leaf-labeled directed acyclic graphs that represent ancestral relationships between species whose past includes non-tree-like events such as hybridization and horizontal gene transfer. Indeed, each phylogenetic network embeds a collection of phylogenetic trees. Referring to the collection of trees that a given phylogenetic network $N$ embeds as the display set of $N$, several questions in the context of the display set of $N$ have recently been analyzed. For example, the widely studied Tree-Containment problem asks if a given phylogenetic tree is contained in the display set of a given network. The focus of this paper are two questions that naturally arise in comparing the display sets of two phylogenetic networks. First, we analyze the problem of deciding if the display sets of two phylogenetic networks have a tree in common. Surprisingly, this problem turns out to be NP-complete even for two temporal normal networks. Second, we investigate the question of whether or not the display sets of two phylogenetic networks are equal. While we recently showed that this problem is polynomial-time solvable for a normal and a tree-child network, it is computationally hard in the general case. In establishing hardness, we show that the problem is contained in the second level of the polynomial-time hierarchy. Specifically, it is $\Pi_2^P$-complete. Along the way, we show that two other problems are also $\Pi_2^P$-complete, one of which being a generalization of Tree-Containment.