Compatible split systems on a multiset

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.

16 pages, 4 figures