Annals of Combinatorics

, Volume 1, Issue 1, pp 339–352

Some variations on a theme by Buneman


  • A. Dress
    • FSPM-StrukturbildugsprozesseUniversity of Bielefeld
  • K. Huber
    • FSPM-StrukturbildugsprozesseUniversity of Bielefeld
  • V. Moulton
    • FSPM-StrukturbildugsprozesseUniversity of Bielefeld

DOI: 10.1007/BF02558485

Cite this article as:
Dress, A., Huber, K. & Moulton, V. Annals of Combinatorics (1997) 1: 339. doi:10.1007/BF02558485


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.

AMS Subject Classification



phylogenetic networksphylogenetic treesmedian networkshypercubeshierarchical clusteringsplit systemssplit decompositionBuneman graphscompatibility of Taxonomic characters
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© Springer-Verlag 1997