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Antipodal metrics and split systems
Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Faculty of Science and Technology, Biology, The Linnaeus Centre for Bioinformatics. V. (Moulton)
2002 (English)In: European Journal of Combinatorics, ISSN 0195-6698, Vol. 23, no 2, 187-200 p.Article in journal (Refereed) Published
Abstract [en]

Recall that a metric d on a finite set X is called antipodal if there exists a map sigma : X --> X: x --> (x) over bar so that d(x, (x) over bar) = d(x, y) + d(y, (x) over bar) holds for all x, y epsilon X. Antipodal metrics canonically arise as metrics induced on specific weighted graphs, although their abundance becomes clearer in light of the fact that any finite metric space can be isometrically embedded in a more or less canonical way into an antipodal metric space called its full antipodal extension. In this paper, we examine in some detail antipodal metrics that are, in addition, totally split decomposable. In particular, we give an explicit characterization of such metrics, and prove that-somewhat surprisingly-the full antipodal extension of a proper metric d on a finite set X is totally split decomposable if and only if d is linear or #X = 3 holds.

Place, publisher, year, edition, pages
2002. Vol. 23, no 2, 187-200 p.
Keyword [en]
decomposition, graphs
National Category
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-71818OAI: oai:DiVA.org:uu-71818DiVA: diva2:99729
Available from: 2005-05-12 Created: 2005-05-12 Last updated: 2011-01-13

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