Representing Partitions on Trees

TL;DR

This paper explores the relationship between partitions and bipartitions in phylogenetic trees, characterizing when certain partition sets correspond to given bipartition multisets and analyzing the computational complexity of these mappings.

Contribution

It introduces the set P(Σ) of partitions corresponding to a multiset of bipartitions, characterizes its non-emptiness, and studies maximum and minimum size partitions, revealing NP-completeness results.

Findings

Characterized when P(Σ) is non-empty.

Identified maximum and minimum size partitions in P(Σ).

Proved NP-completeness of deciding non-emptiness of P(Σ).

Abstract

In evolutionary biology, biologists often face the problem of constructing a phylogenetic tree on a set $X$ of species from a multiset $\Pi$ of partitions corresponding to various attributes of these species. One approach that is used to solve this problem is to try instead to associate a tree (or even a network) to the multiset $\Sigma_{\Pi}$ consisting of all those bipartitions $\{A,X-A\}$ with $A$ a part of some partition in $\Pi$. The rational behind this approach is that a phylogenetic tree with leaf set $X$ can be uniquely represented by the set of bipartitions of $X$ induced by its edges. Motivated by these considerations, given a multiset $\Sigma$ of bipartitions corresponding to a phylogenetic tree on $X$, in this paper we introduce and study the set $P(\Sigma)$ consisting of those multisets of partitions $\Pi$ of $X$ with $\Sigma_{\Pi}=\Sigma$. More specifically, we characterize when $P(\Sigma)$ is non-empty, and also identify some partitions in $P(\Sigma)$ that are of maximum and minimum size. We also show that it is NP-complete to decide when $P(\Sigma)$ is non-empty in case $\Sigma$ is an arbitrary multiset of bipartitions of $X$. Ultimately, we hope that by gaining a better understanding of the mapping that takes an arbitrary partition system $\Pi$ to the multiset $\Sigma_{\Pi}$, we will obtain new insights into the use of median networks and, more generally, split-networks to visualize sets of partitions.