Compatible Split Systems on a Multiset

Abstract

A split system on a multiset $\mathcal M$ is a multiset 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. Compatibility of split systems is a key concept in phylogenetics, and compatible split systems have applications to, for example, multi-labelled phylogenetic trees. In this contribution, we present a novel characterization for compatible split systems, and for split systems admitting a unique representation by a tree. In addition, we show that a conjecture on compatibility stated in 2008 holds for some large classes of split systems.