uu.seUppsala University Publications

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Concerning the relationship between realizations and tight spans of finite metricsPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 2007 (English)In: Discrete & Computational Geometry, ISSN 0179-5376, E-ISSN 1432-0444, Vol. 38, no 3, p. 605-614Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

2007. Vol. 38, no 3, p. 605-614
##### National Category

Mathematics
##### Identifiers

URN: urn:nbn:se:uu:diva-96396DOI: 10.1007/s00454-007-1352-5ISI: 000249696700007OAI: oai:DiVA.org:uu-96396DiVA, id: diva2:170956
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Available from: 2007-11-07 Created: 2007-11-07 Last updated: 2017-12-14Bibliographically approved
##### In thesis

Given a metric d on a finite set X, a realization of d is a weighted graph $G=(V,E,w\colon \ E \to {\Bbb R}_{>0})$ with $X \subseteq V$ such that for all $x,y \in X$ the length of any shortest path in G between x and y equals d(x,y). In this paper we consider two special kinds of realizations, optimal realizations and hereditarily optimal realizations, and their relationship with the so-called tight span. In particular, we present an infinite family of metrics {d_{k}}_{k≥1}, and—using a new characterization for when the so-called underlying graph of a metric is an optimal realization that we also present—we prove that d_{k} has (as a function of k) exponentially many optimal realizations with distinct degree sequences. We then show that this family of metrics provides counter-examples to a conjecture made by Dress in 1984 concerning the relationship between optimal realizations and the tight span, and a negative reply to a question posed by Althofer in 1988 on the relationship between optimal and hereditarily optimal realizations.

1. Optimal and Hereditarily Optimal Realizations of Metric Spaces$(function(){PrimeFaces.cw("OverlayPanel","overlay170958",{id:"formSmash:j_idt892:0:j_idt899",widgetVar:"overlay170958",target:"formSmash:j_idt892:0:parentLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

doi
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