In 1971, Peter Buneman proposed a way to construct a tree from a collection of pairwise compatible splits. This construction immediately generalizes to arbitrary collections of splits, and yields a connected median graph, called the Buneman graph. In this paper, we prove that the vertices and the edges of this graph can be described in a very simple way: given a collection of splitsS, the vertices of the Buneman graph correspond precisely to the subsetsS′ ofS such that the splits inS′ are pairwise incompatible and the edges correspond to pairs (S′, S) withS′ as above andS∈S′. Using this characterization, it is much more straightforward to construct the vertices of the Buneman graph than using prior constructions. We also recover as an immediate consequence of this enumeration that the Buneman graph is a tree, that is, that the number of vertices exceeds the number of edges (by one), if and only if any two distinct splits inS are compatible.

In 1971, Peter Buneman presented a paper in which he described, amongst other things, a way to construct a tree from a collection of pairwise compatible splits of a finite set. This construction is immediately generalizable to a collection of arbitrary splits, in which case it gives rise to theBuneman graph, a certain connected, median graph representing the given collection of splits in a canonical fashion. In this paper, we look at the complex obtained by filling those faces of the associated hypercube whose vertices consist only of vertices in the Buneman graph, a complex that we call theBuneman complex. In particular, we give a natural filtration of the Buneman complex, and show that this filtration collapses when the system of splitis is weakly compatible. In the case where the family of splits is not weakly compatible, we also see that the filtration naturally gives us a way to associate a graded hierarchy of phylogenetic networks to the collection of splits.