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Representing Partitions on Trees

In evolutionary biology, biologists often face the problem of constructing a phylogenetic tree on a set $X$ of species from a multiset $Π$ 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 $Σ_Π$ consisting of all those bipartitions $\{A,X-A\}$ with $A$ a part of some partition in $Π$. 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 $Σ$ of bipartitions corresponding to a phylogenetic tree on $X$, in this paper we introduce and study the set $P(Σ)$ consisting of those multisets of partitions $Π$ of $X$ with $Σ_Π=Σ$. More specifically, we characterize when $P(Σ)$ is non-empty, and also identify some partitions in $P(Σ)$ that are of maximum and minimum size. We also show that it is NP-complete to decide when $P(Σ)$ is non-empty in case $Σ$ 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 $Π$ to the multiset $Σ_Π$, we will obtain new insights into the use of median networks and, more generally, split-networks to visualize sets of partitions.