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PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 2016 (English)In: Israel Journal of Mathematics, ISSN 0021-2172, E-ISSN 1565-8511, Vol. 212, no 1, p. 37-79Article in journal (Refereed) Published
##### Resource type

Text
##### Abstract [en]

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

Springer-Verlag New York, 2016. Vol. 212, no 1, p. 37-79
##### National Category

Mathematics
##### Identifiers

URN: urn:nbn:se:kth:diva-189397DOI: 10.1007/s11856-016-1294-9ISI: 000377265600002Scopus ID: 2-s2.0-84971014023OAI: oai:DiVA.org:kth-189397DiVA: diva2:948162
#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt434",{id:"formSmash:j_idt434",widgetVar:"widget_formSmash_j_idt434",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt440",{id:"formSmash:j_idt440",widgetVar:"widget_formSmash_j_idt440",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt446",{id:"formSmash:j_idt446",widgetVar:"widget_formSmash_j_idt446",multiple:true});
##### Note

We consider two systems (alpha(1), ... ,alpha(m)) and (beta(1), ... , beta(n)) of simple curves drawn on a compact two-dimensional surface M with boundary. Each alpha(i) and each beta(j) is either an arc meeting the boundary of M at its two endpoints, or a closed curve. The a, are pairwise disjoint except for possibly sharing endpoints, and similarly for the beta(j). We want to "untangle" the beta(j) from the alpha(i) by a self-homeomorphism of M; more precisely, we seek a homeomorphism phi: M -> M fixing the boundary of M pointwise such that the total number of crossings of the a, with the phi(beta(j)) is as small as possible. This problem is motivated by an application in the algorithmic theory of embeddings and 3 -manifolds. We prove that if M is planar, i.e., a sphere with h >= 0 boundary components ("holes"), then O(mn) crossings can be achieved (independently of h), which is asymptotically tight, as an easy lower bound shows. In general, for an arbitrary (orientable or nonorientable) surface M with h holes and of (orientable or nonorientable) genus g >= 0, we obtain an O((m + n)(4)) upper bound, again independent of h and g. The proofs rely, among other things, on a result concerning simultaneous planar drawings of graphs by Erten and Kobourov.

QC 20160708

Available from: 2016-07-08 Created: 2016-07-04 Last updated: 2017-11-28Bibliographically approved
doi
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CiteExport$(function(){PrimeFaces.cw("TieredMenu","widget_formSmash_lower_j_idt1196",{id:"formSmash:lower:j_idt1196",widgetVar:"widget_formSmash_lower_j_idt1196",autoDisplay:true,overlay:true,my:"left top",at:"left bottom",trigger:"formSmash:lower:exportLink",triggerEvent:"click"});}); $(function(){PrimeFaces.cw("OverlayPanel","widget_formSmash_lower_j_idt1197_j_idt1199",{id:"formSmash:lower:j_idt1197:j_idt1199",widgetVar:"widget_formSmash_lower_j_idt1197_j_idt1199",target:"formSmash:lower:j_idt1197:permLink",showEffect:"blind",hideEffect:"fade",my:"right top",at:"right bottom",showCloseIcon:true});});