New Classes of Counterexamples to Hendrickson’s Global Rigidity Conjecture
We examine the generic local and global rigidity of various graphs in ℝ d . Bruce Hendrickson showed that some necessary conditions for generic global rigidity are ( d +1)-connectedness and generic redundant rigidity, and hypothesized that they were sufficient in all dimensions. We analyze two class...
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| Veröffentlicht in: | Discrete & computational geometry 2011-04, Vol.45 (3), p.574-591 |
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| container_title | Discrete & computational geometry |
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| creator | Frank, Samuel Jiang, Jiayang |
| description | We examine the generic local and global rigidity of various graphs in ℝ
d
. Bruce Hendrickson showed that some necessary conditions for generic global rigidity are (
d
+1)-connectedness and generic redundant rigidity, and hypothesized that they were sufficient in all dimensions. We analyze two classes of graphs that satisfy Hendrickson’s conditions for generic global rigidity, yet fail to be generically globally rigid. We find a large family of bipartite graphs for
d
>3, and we define a construction that generates infinitely many graphs in ℝ
5
. Finally, we state some conjectures for further exploration. |
| doi_str_mv | 10.1007/s00454-010-9259-y |
| format | Article |
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d
. Bruce Hendrickson showed that some necessary conditions for generic global rigidity are (
d
+1)-connectedness and generic redundant rigidity, and hypothesized that they were sufficient in all dimensions. We analyze two classes of graphs that satisfy Hendrickson’s conditions for generic global rigidity, yet fail to be generically globally rigid. We find a large family of bipartite graphs for
d
>3, and we define a construction that generates infinitely many graphs in ℝ
5
. Finally, we state some conjectures for further exploration.</description><identifier>ISSN: 0179-5376</identifier><identifier>EISSN: 1432-0444</identifier><identifier>DOI: 10.1007/s00454-010-9259-y</identifier><identifier>CODEN: DCGEER</identifier><language>eng</language><publisher>New York: Springer-Verlag</publisher><subject>Combinatorics ; Computational geometry ; Computational Mathematics and Numerical Analysis ; Construction ; Euclidean space ; Exploration ; Geometry ; Graphs ; Mathematics ; Mathematics and Statistics ; Matrix ; Redundant ; Rigidity</subject><ispartof>Discrete & computational geometry, 2011-04, Vol.45 (3), p.574-591</ispartof><rights>Springer Science+Business Media, LLC 2010</rights><rights>Springer Science+Business Media, LLC 2011</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c456t-beb4c2e593ef919b563fc3d83386b011c536de7475d7705b5358ed5a037e641c3</citedby><cites>FETCH-LOGICAL-c456t-beb4c2e593ef919b563fc3d83386b011c536de7475d7705b5358ed5a037e641c3</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/s00454-010-9259-y$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00454-010-9259-y$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27924,27925,41488,42557,51319</link.rule.ids></links><search><creatorcontrib>Frank, Samuel</creatorcontrib><creatorcontrib>Jiang, Jiayang</creatorcontrib><title>New Classes of Counterexamples to Hendrickson’s Global Rigidity Conjecture</title><title>Discrete & computational geometry</title><addtitle>Discrete Comput Geom</addtitle><description>We examine the generic local and global rigidity of various graphs in ℝ
d
. Bruce Hendrickson showed that some necessary conditions for generic global rigidity are (
d
+1)-connectedness and generic redundant rigidity, and hypothesized that they were sufficient in all dimensions. We analyze two classes of graphs that satisfy Hendrickson’s conditions for generic global rigidity, yet fail to be generically globally rigid. We find a large family of bipartite graphs for
d
>3, and we define a construction that generates infinitely many graphs in ℝ
5
. Finally, we state some conjectures for further exploration.</description><subject>Combinatorics</subject><subject>Computational geometry</subject><subject>Computational Mathematics and Numerical Analysis</subject><subject>Construction</subject><subject>Euclidean space</subject><subject>Exploration</subject><subject>Geometry</subject><subject>Graphs</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Matrix</subject><subject>Redundant</subject><subject>Rigidity</subject><issn>0179-5376</issn><issn>1432-0444</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2011</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><recordid>eNp1kM1KxDAQgIMouK4-gLfixVN10vy1Rym6KywKoufQplPp2m3WpEV78zV8PZ_ELBUEwdPA8H3D8BFySuGCAqhLD8AFj4FCnCUii8c9MqOcJTFwzvfJDKjKYsGUPCRH3q8h4BmkM7K6w7cobwvv0Ue2jnI7dD06fC822zasehstsatcY1687b4-Pn20aG1ZtNFD89xUTT8GpVuj6QeHx-SgLlqPJz9zTp5urh_zZby6X9zmV6vYcCH7uMSSmwRFxrDOaFYKyWrDqpSxVJZAqRFMVqi4EpVSIErBRIqVKIAplJwaNifn092ts68D-l5vGm-wbYsO7eB1KqmQAlgayLM_5NoOrgvP6VQAZwo4BIhOkHHWe4e13rpmU7hRU9C7unqqq0Ndvaurx-Akk-MD2z2j-z38v_QNrm19mg</recordid><startdate>20110401</startdate><enddate>20110401</enddate><creator>Frank, Samuel</creator><creator>Jiang, Jiayang</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>PTHSS</scope><scope>Q9U</scope></search><sort><creationdate>20110401</creationdate><title>New Classes of Counterexamples to Hendrickson’s Global Rigidity Conjecture</title><author>Frank, Samuel ; 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d
. Bruce Hendrickson showed that some necessary conditions for generic global rigidity are (
d
+1)-connectedness and generic redundant rigidity, and hypothesized that they were sufficient in all dimensions. We analyze two classes of graphs that satisfy Hendrickson’s conditions for generic global rigidity, yet fail to be generically globally rigid. We find a large family of bipartite graphs for
d
>3, and we define a construction that generates infinitely many graphs in ℝ
5
. Finally, we state some conjectures for further exploration.</abstract><cop>New York</cop><pub>Springer-Verlag</pub><doi>10.1007/s00454-010-9259-y</doi><tpages>18</tpages><oa>free_for_read</oa></addata></record> |
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| ispartof | Discrete & computational geometry, 2011-04, Vol.45 (3), p.574-591 |
| issn | 0179-5376 1432-0444 |
| language | eng |
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| source | SpringerLink Journals - AutoHoldings |
| subjects | Combinatorics Computational geometry Computational Mathematics and Numerical Analysis Construction Euclidean space Exploration Geometry Graphs Mathematics Mathematics and Statistics Matrix Redundant Rigidity |
| title | New Classes of Counterexamples to Hendrickson’s Global Rigidity Conjecture |
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