Annals of Combinatorics 2(1998) 299-311

A Comparison Between Two Distinct Continuous Models in Projective Cluster Theory: The Median and the Tight--Span Construction

A.W.M. Dress1, K.T. Huber2, and V. Moulton 3

1FSPM--Strukturbildungsprozesse, University of Bielefeld, Bielefeld, D--33501, Germany

2Institute of Fundamental Sciences, Massey University, Private Bag 11 222, Palmerston North New Zealand

3FMI, Mid Sweden University, Sundsvall, S 851-70, Sweden}

Received August 20, 1998

AMS Subject Classification: 04A03, 04A20, 05C99, 52B99, 92B99

Abstract. It is possible to consider two variants of cluster theory: In affine cluster theory, one considers collections of subsets of a given set X of objects or states, whereas in projective cluster theoty, one considers collections of splits (or bipartitions) of that set. In both contexts, it can be desirable to produce a continuous model, that is, a space T encompassing the given set X which represents in a well-specified and more or less parsimonious way all possible intermediate objects or transition statescompatible with certain restrictions derived from the given collection of subsets or splits. We investigate an interesting and intriguing relationship between two such constructions that appear in the context of projective cluster theory: The Buneman construction and the tight-span(or just T) construction.

Keywords: Buneman graph, cluster theory, split systems, split decomposition, T-theory, T-construction, pairwise compatibility, weak compatibility, median networks, hypercube, phylogenetic trees, phylogenetic networks

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