A Polynomial Bound for Untangling Geometric Planar Graphs

To untangle a geometric graph means to move some of the vertices so that the resulting geometric graph has no crossings. Pach and Tardos (Discrete Comput. Geom. 28(4): 585–592, 2002 ) asked if every n -vertex geometric planar graph can be untangled while keeping at least n ε vertices fixed. We answe...

Ausführliche Beschreibung

Gespeichert in:
Bibliographische Detailangaben
Veröffentlicht in:Discrete & computational geometry 2009-12, Vol.42 (4), p.570-585
Hauptverfasser: Bose, Prosenjit, Dujmović, Vida, Hurtado, Ferran, Langerman, Stefan, Morin, Pat, Wood, David R.
Format: Artikel
Sprache:eng
Schlagworte:
Online-Zugang:Volltext
Tags: Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
container_end_page 585
container_issue 4
container_start_page 570
container_title Discrete & computational geometry
container_volume 42
creator Bose, Prosenjit
Dujmović, Vida
Hurtado, Ferran
Langerman, Stefan
Morin, Pat
Wood, David R.
description To untangle a geometric graph means to move some of the vertices so that the resulting geometric graph has no crossings. Pach and Tardos (Discrete Comput. Geom. 28(4): 585–592, 2002 ) asked if every n -vertex geometric planar graph can be untangled while keeping at least n ε vertices fixed. We answer this question in the affirmative with ε =1/4. The previous best known bound was . We also consider untangling geometric trees. It is known that every n -vertex geometric tree can be untangled while keeping at least vertices fixed, while the best upper bound was . We answer a question of Spillner and Wolff ( http://arxiv.org/abs/0709.0170 ) by closing this gap for untangling trees. In particular, we show that for infinitely many values of  n , there is an n -vertex geometric tree that cannot be untangled while keeping more than vertices fixed.
doi_str_mv 10.1007/s00454-008-9125-3
format Article
fullrecord <record><control><sourceid>proquest_csuc_</sourceid><recordid>TN_cdi_csuc_recercat_oai_recercat_cat_2072_191653</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>1917123281</sourcerecordid><originalsourceid>FETCH-LOGICAL-c466t-8aaa938fa8df5b2591ff46c444c351afe7b97cd2af877b5a10d314ab5b5145693</originalsourceid><addsrcrecordid>eNp1kE1OwzAQRi0EEqVwAHYR-4B_45hdqaAgIdEFXVsTxy6pUrvYyaK34SycjERFohsWo9FI33safQhdE3xLMJZ3CWMueI5xmStCRc5O0IRwRnPMOT9FE0ykygWTxTm6SGmDh7jC5QTdz7JlaPc-bBtos4fQ-zpzIWYr34Fft41fZwsbtraLjcmWLXiI31-LCLuPdInOHLTJXv3uKVo9Pb7Pn_PXt8XLfPaaG14UXV4CgGKlg7J2oqJCEed4YYa3DBMEnJWVkqam4EopKwEE14xwqEQlCBeFYlNEDl6TeqOjNTYa6HSA5u8Yh2JJNVGkEGxgbg7MLobP3qZOb0If_fDmkCgkEyWjR-IYUorW6V1sthD3mmA9tqoPreqhVT22qkcxPTBpyPq1jUfif6Efzyp5pQ</addsrcrecordid><sourcetype>Open Access Repository</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>196735832</pqid></control><display><type>article</type><title>A Polynomial Bound for Untangling Geometric Planar Graphs</title><source>SpringerNature Journals</source><source>Recercat</source><creator>Bose, Prosenjit ; Dujmović, Vida ; Hurtado, Ferran ; Langerman, Stefan ; Morin, Pat ; Wood, David R.</creator><creatorcontrib>Bose, Prosenjit ; Dujmović, Vida ; Hurtado, Ferran ; Langerman, Stefan ; Morin, Pat ; Wood, David R.</creatorcontrib><description>To untangle a geometric graph means to move some of the vertices so that the resulting geometric graph has no crossings. Pach and Tardos (Discrete Comput. Geom. 28(4): 585–592, 2002 ) asked if every n -vertex geometric planar graph can be untangled while keeping at least n ε vertices fixed. We answer this question in the affirmative with ε =1/4. The previous best known bound was . We also consider untangling geometric trees. It is known that every n -vertex geometric tree can be untangled while keeping at least vertices fixed, while the best upper bound was . We answer a question of Spillner and Wolff ( http://arxiv.org/abs/0709.0170 ) by closing this gap for untangling trees. In particular, we show that for infinitely many values of  n , there is an n -vertex geometric tree that cannot be untangled while keeping more than vertices fixed.</description><identifier>ISSN: 0179-5376</identifier><identifier>EISSN: 1432-0444</identifier><identifier>DOI: 10.1007/s00454-008-9125-3</identifier><identifier>CODEN: DCGEER</identifier><language>eng</language><publisher>New York: Springer-Verlag</publisher><subject>Combinatorics ; Computational Mathematics and Numerical Analysis ; Crossings ; Discrete geometry ; Geometria ; Geometria convexa i discreta ; Geometria discreta ; Grafs, Teoria de ; Graph theory ; Matemàtiques i estadística ; Mathematics ; Mathematics and Statistics ; Polinomis ; Polynomials ; Àrees temàtiques de la UPC</subject><ispartof>Discrete &amp; computational geometry, 2009-12, Vol.42 (4), p.570-585</ispartof><rights>Springer Science+Business Media, LLC 2008</rights><rights>Springer Science+Business Media, LLC 2009</rights><rights>Attribution-NonCommercial-NoDerivs 3.0 Spain info:eu-repo/semantics/openAccess &lt;a href="http://creativecommons.org/licenses/by-nc-nd/3.0/es/"&gt;http://creativecommons.org/licenses/by-nc-nd/3.0/es/&lt;/a&gt;</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c466t-8aaa938fa8df5b2591ff46c444c351afe7b97cd2af877b5a10d314ab5b5145693</citedby><cites>FETCH-LOGICAL-c466t-8aaa938fa8df5b2591ff46c444c351afe7b97cd2af877b5a10d314ab5b5145693</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s00454-008-9125-3$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00454-008-9125-3$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>230,314,780,784,885,26974,27924,27925,41488,42557,51319</link.rule.ids></links><search><creatorcontrib>Bose, Prosenjit</creatorcontrib><creatorcontrib>Dujmović, Vida</creatorcontrib><creatorcontrib>Hurtado, Ferran</creatorcontrib><creatorcontrib>Langerman, Stefan</creatorcontrib><creatorcontrib>Morin, Pat</creatorcontrib><creatorcontrib>Wood, David R.</creatorcontrib><title>A Polynomial Bound for Untangling Geometric Planar Graphs</title><title>Discrete &amp; computational geometry</title><addtitle>Discrete Comput Geom</addtitle><description>To untangle a geometric graph means to move some of the vertices so that the resulting geometric graph has no crossings. Pach and Tardos (Discrete Comput. Geom. 28(4): 585–592, 2002 ) asked if every n -vertex geometric planar graph can be untangled while keeping at least n ε vertices fixed. We answer this question in the affirmative with ε =1/4. The previous best known bound was . We also consider untangling geometric trees. It is known that every n -vertex geometric tree can be untangled while keeping at least vertices fixed, while the best upper bound was . We answer a question of Spillner and Wolff ( http://arxiv.org/abs/0709.0170 ) by closing this gap for untangling trees. In particular, we show that for infinitely many values of  n , there is an n -vertex geometric tree that cannot be untangled while keeping more than vertices fixed.</description><subject>Combinatorics</subject><subject>Computational Mathematics and Numerical Analysis</subject><subject>Crossings</subject><subject>Discrete geometry</subject><subject>Geometria</subject><subject>Geometria convexa i discreta</subject><subject>Geometria discreta</subject><subject>Grafs, Teoria de</subject><subject>Graph theory</subject><subject>Matemàtiques i estadística</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Polinomis</subject><subject>Polynomials</subject><subject>Àrees temàtiques de la UPC</subject><issn>0179-5376</issn><issn>1432-0444</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2009</creationdate><recordtype>article</recordtype><sourceid>8G5</sourceid><sourceid>ABUWG</sourceid><sourceid>AFKRA</sourceid><sourceid>AZQEC</sourceid><sourceid>BENPR</sourceid><sourceid>CCPQU</sourceid><sourceid>DWQXO</sourceid><sourceid>GNUQQ</sourceid><sourceid>GUQSH</sourceid><sourceid>M2O</sourceid><sourceid>XX2</sourceid><recordid>eNp1kE1OwzAQRi0EEqVwAHYR-4B_45hdqaAgIdEFXVsTxy6pUrvYyaK34SycjERFohsWo9FI33safQhdE3xLMJZ3CWMueI5xmStCRc5O0IRwRnPMOT9FE0ykygWTxTm6SGmDh7jC5QTdz7JlaPc-bBtos4fQ-zpzIWYr34Fft41fZwsbtraLjcmWLXiI31-LCLuPdInOHLTJXv3uKVo9Pb7Pn_PXt8XLfPaaG14UXV4CgGKlg7J2oqJCEed4YYa3DBMEnJWVkqam4EopKwEE14xwqEQlCBeFYlNEDl6TeqOjNTYa6HSA5u8Yh2JJNVGkEGxgbg7MLobP3qZOb0If_fDmkCgkEyWjR-IYUorW6V1sthD3mmA9tqoPreqhVT22qkcxPTBpyPq1jUfif6Efzyp5pQ</recordid><startdate>20091201</startdate><enddate>20091201</enddate><creator>Bose, Prosenjit</creator><creator>Dujmović, Vida</creator><creator>Hurtado, Ferran</creator><creator>Langerman, Stefan</creator><creator>Morin, Pat</creator><creator>Wood, David R.</creator><general>Springer-Verlag</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><scope>3V.</scope><scope>7SC</scope><scope>7TB</scope><scope>7XB</scope><scope>88I</scope><scope>8AL</scope><scope>8AO</scope><scope>8FD</scope><scope>8FE</scope><scope>8FG</scope><scope>8FK</scope><scope>8G5</scope><scope>ABJCF</scope><scope>ABUWG</scope><scope>AFKRA</scope><scope>ARAPS</scope><scope>AZQEC</scope><scope>BENPR</scope><scope>BGLVJ</scope><scope>CCPQU</scope><scope>DWQXO</scope><scope>FR3</scope><scope>GNUQQ</scope><scope>GUQSH</scope><scope>HCIFZ</scope><scope>JQ2</scope><scope>K7-</scope><scope>KR7</scope><scope>L6V</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope><scope>M0N</scope><scope>M2O</scope><scope>M2P</scope><scope>M7S</scope><scope>MBDVC</scope><scope>P5Z</scope><scope>P62</scope><scope>PADUT</scope><scope>PQEST</scope><scope>PQQKQ</scope><scope>PQUKI</scope><scope>PRINS</scope><scope>PTHSS</scope><scope>Q9U</scope><scope>XX2</scope></search><sort><creationdate>20091201</creationdate><title>A Polynomial Bound for Untangling Geometric Planar Graphs</title><author>Bose, Prosenjit ; Dujmović, Vida ; Hurtado, Ferran ; Langerman, Stefan ; Morin, Pat ; Wood, David R.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c466t-8aaa938fa8df5b2591ff46c444c351afe7b97cd2af877b5a10d314ab5b5145693</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2009</creationdate><topic>Combinatorics</topic><topic>Computational Mathematics and Numerical Analysis</topic><topic>Crossings</topic><topic>Discrete geometry</topic><topic>Geometria</topic><topic>Geometria convexa i discreta</topic><topic>Geometria discreta</topic><topic>Grafs, Teoria de</topic><topic>Graph theory</topic><topic>Matemàtiques i estadística</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Polinomis</topic><topic>Polynomials</topic><topic>Àrees temàtiques de la UPC</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Bose, Prosenjit</creatorcontrib><creatorcontrib>Dujmović, Vida</creatorcontrib><creatorcontrib>Hurtado, Ferran</creatorcontrib><creatorcontrib>Langerman, Stefan</creatorcontrib><creatorcontrib>Morin, Pat</creatorcontrib><creatorcontrib>Wood, David R.</creatorcontrib><collection>CrossRef</collection><collection>ProQuest Central (Corporate)</collection><collection>Computer and Information Systems Abstracts</collection><collection>Mechanical &amp; Transportation Engineering Abstracts</collection><collection>ProQuest Central (purchase pre-March 2016)</collection><collection>Science Database (Alumni Edition)</collection><collection>Computing Database (Alumni Edition)</collection><collection>ProQuest Pharma Collection</collection><collection>Technology Research Database</collection><collection>ProQuest SciTech Collection</collection><collection>ProQuest Technology Collection</collection><collection>ProQuest Central (Alumni) (purchase pre-March 2016)</collection><collection>Research Library (Alumni Edition)</collection><collection>Materials Science &amp; Engineering Collection</collection><collection>ProQuest Central (Alumni Edition)</collection><collection>ProQuest Central UK/Ireland</collection><collection>Advanced Technologies &amp; Aerospace Collection</collection><collection>ProQuest Central Essentials</collection><collection>ProQuest Central</collection><collection>Technology Collection</collection><collection>ProQuest One Community College</collection><collection>ProQuest Central Korea</collection><collection>Engineering Research Database</collection><collection>ProQuest Central Student</collection><collection>Research Library Prep</collection><collection>SciTech Premium Collection</collection><collection>ProQuest Computer Science Collection</collection><collection>Computer Science Database</collection><collection>Civil Engineering Abstracts</collection><collection>ProQuest Engineering Collection</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts – Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><collection>Computing Database</collection><collection>Research Library</collection><collection>Science Database</collection><collection>Engineering Database</collection><collection>Research Library (Corporate)</collection><collection>Advanced Technologies &amp; Aerospace Database</collection><collection>ProQuest Advanced Technologies &amp; Aerospace Collection</collection><collection>Research Library China</collection><collection>ProQuest One Academic Eastern Edition (DO NOT USE)</collection><collection>ProQuest One Academic</collection><collection>ProQuest One Academic UKI Edition</collection><collection>ProQuest Central China</collection><collection>Engineering Collection</collection><collection>ProQuest Central Basic</collection><collection>Recercat</collection><jtitle>Discrete &amp; computational geometry</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Bose, Prosenjit</au><au>Dujmović, Vida</au><au>Hurtado, Ferran</au><au>Langerman, Stefan</au><au>Morin, Pat</au><au>Wood, David R.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>A Polynomial Bound for Untangling Geometric Planar Graphs</atitle><jtitle>Discrete &amp; computational geometry</jtitle><stitle>Discrete Comput Geom</stitle><date>2009-12-01</date><risdate>2009</risdate><volume>42</volume><issue>4</issue><spage>570</spage><epage>585</epage><pages>570-585</pages><issn>0179-5376</issn><eissn>1432-0444</eissn><coden>DCGEER</coden><abstract>To untangle a geometric graph means to move some of the vertices so that the resulting geometric graph has no crossings. Pach and Tardos (Discrete Comput. Geom. 28(4): 585–592, 2002 ) asked if every n -vertex geometric planar graph can be untangled while keeping at least n ε vertices fixed. We answer this question in the affirmative with ε =1/4. The previous best known bound was . We also consider untangling geometric trees. It is known that every n -vertex geometric tree can be untangled while keeping at least vertices fixed, while the best upper bound was . We answer a question of Spillner and Wolff ( http://arxiv.org/abs/0709.0170 ) by closing this gap for untangling trees. In particular, we show that for infinitely many values of  n , there is an n -vertex geometric tree that cannot be untangled while keeping more than vertices fixed.</abstract><cop>New York</cop><pub>Springer-Verlag</pub><doi>10.1007/s00454-008-9125-3</doi><tpages>16</tpages><oa>free_for_read</oa></addata></record>
fulltext fulltext
identifier ISSN: 0179-5376
ispartof Discrete & computational geometry, 2009-12, Vol.42 (4), p.570-585
issn 0179-5376
1432-0444
language eng
recordid cdi_csuc_recercat_oai_recercat_cat_2072_191653
source SpringerNature Journals; Recercat
subjects Combinatorics
Computational Mathematics and Numerical Analysis
Crossings
Discrete geometry
Geometria
Geometria convexa i discreta
Geometria discreta
Grafs, Teoria de
Graph theory
Matemàtiques i estadística
Mathematics
Mathematics and Statistics
Polinomis
Polynomials
Àrees temàtiques de la UPC
title A Polynomial Bound for Untangling Geometric Planar Graphs
url https://sfx.bib-bvb.de/sfx_tum?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2024-12-22T15%3A44%3A40IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-proquest_csuc_&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=A%20Polynomial%20Bound%20for%20Untangling%20Geometric%20Planar%C2%A0Graphs&rft.jtitle=Discrete%20&%20computational%20geometry&rft.au=Bose,%20Prosenjit&rft.date=2009-12-01&rft.volume=42&rft.issue=4&rft.spage=570&rft.epage=585&rft.pages=570-585&rft.issn=0179-5376&rft.eissn=1432-0444&rft.coden=DCGEER&rft_id=info:doi/10.1007/s00454-008-9125-3&rft_dat=%3Cproquest_csuc_%3E1917123281%3C/proquest_csuc_%3E%3Curl%3E%3C/url%3E&disable_directlink=true&sfx.directlink=off&sfx.report_link=0&rft_id=info:oai/&rft_pqid=196735832&rft_id=info:pmid/&rfr_iscdi=true