Some variations on a theme by Buneman

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.