Compatible split systems on a multiset

TL;DR

This paper introduces a new characterization of compatible split systems on multisets, providing insights into their tree representations and confirming a 2008 conjecture for specific classes.

Contribution

It offers a novel characterization of compatible split systems and proves a longstanding conjecture for certain classes of these systems.

Findings

New characterization of compatible split systems

Unique tree representation criteria established

Verification of a 2008 conjecture for specific classes

Abstract

A split system on a multiset $\mathcal M$ is a set of bipartitions of $\mathcal M$. Such a split system $\mathfrak S$ is compatible if it can be represented by a tree in such a way that the vertices of the tree are labelled by the elements in $\mathcal M$, the removal of each edge in the tree yields a bipartition in $\mathfrak S$ by taking the labels of the two resulting components, and every bipartition in $\mathfrak S$ can be obtained from the tree in this way. In this contribution, we present a novel characterization for compatible split systems, and for split systems admitting a unique tree representation. In addition, we show that a conjecture on compatibility stated in 2008 holds for some large classes of split systems.