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 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 rationale 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 nonempty 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 nonempty in the case when 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.