Slim Sets of Binary Trees

TL;DR

This paper investigates compatibility conditions for sets of binary phylogenetic trees, providing a sufficient condition for compatibility and a new proof for the minimal cardinality characterization of quartet sets that are uniquely displayed by a single phylogenetic tree.

Contribution

It introduces a new sufficient condition for compatibility of binary phylogenetic trees and offers a concise proof for the minimal cardinality characterization of uniquely displayed quartet sets.

Findings

Proves a new sufficient condition for compatibility of binary phylogenetic trees.

Provides a short, self-contained proof for the minimal cardinality characterization of quartet sets.

Enhances understanding of compatibility and uniqueness in phylogenetic tree sets.

Abstract

A classical problem in phylogenetic tree analysis is to decide whether there is a phylogenetic tree $T$ that contains all information of a given collection $\cP$ of phylogenetic trees. If the answer is "yes" we say that $\cP$ is compatible and $T$ displays $\cP$. This decision problem is NP-complete even if all input trees are quartets, that is binary trees with exactly four leaves. In this paper, we prove a sufficient condition for a set of binary phylogenetic trees to be compatible. That result is used to give a short and self-contained proof of the known characterization of quartet sets of minimal cardinality which are displayed by a unique phylogenetic tree.