On the number of vertices and edges of the Buneman graph

Authors

A. Dress

FSPM-StrukturbildugsprozesseUniversity of Bielefeld

M. Hendy

Mathematics DepartmentMassey University

K. Huber

FSPM-StrukturbildugsprozesseUniversity of Bielefeld

V. Moulton

FSPM-StrukturbildugsprozesseUniversity of Bielefeld

Article

Received:

DOI:
10.1007/BF02558484

Cite this article as:

Dress, A., Hendy, M., Huber, K. et al. Annals of Combinatorics (1997) 1: 329. doi:10.1007/BF02558484

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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

05C9905C0504A0303E0592B99

Keywords

Buneman graphssplit systemssplit decompositonphylogenetic networksphylogenetic treesmedian networkshypercubescompatibility of taxonomic characters