Annals of Combinatorics

, Volume 1, Issue 1, pp 329–337

On the number of vertices and edges of the Buneman graph

  • A. Dress
  • M. Hendy
  • K. Huber
  • V. Moulton
Article

Abstract

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.

AMS Subject Classification

05C99 05C05 04A03 03E05 92B99 

Keywords

Buneman graphs split systems split decompositon phylogenetic networks phylogenetic trees median networks hypercubes compatibility of taxonomic characters 

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Copyright information

© Springer-Verlag 1997

Authors and Affiliations

  • A. Dress
    • 1
  • M. Hendy
    • 2
  • K. Huber
    • 1
  • V. Moulton
    • 1
  1. 1.FSPM-StrukturbildugsprozesseUniversity of BielefeldBielefeldGermany
  2. 2.Mathematics DepartmentMassey UniversityPalmerston NorthNew Zealand

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