Combinatorics of Linked Systems of Quartet Trees

TL;DR

This paper investigates the combinatorial properties of linked systems of quartet trees in phylogenetics, establishing bounds on decisiveness and probability, and demonstrating the optimality of these bounds.

Contribution

It introduces a new combinatorial framework for understanding decisiveness in quartet systems and proves the optimality of the bounds on the number of quartets needed.

Findings

Identifies a critical value k for decisive quartet collections

Proves the bound on k is optimal

Provides a lower-bound on the probability of decisiveness

Abstract

We apply classical quartet techniques to the problem of phylogenetic decisiveness and find a value $k$ such that all collections of at least $k$ quartets are decisive. Moreover, we prove that this bound is optimal and give a lower-bound on the probability that a collection of quartets is decisive.