{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:05:04Z","timestamp":1787321104776,"version":"build-2736575974"},"reference-count":15,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","funder":[{"DOI":"10.13039\/501100004543","name":"China Scholarship Council","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100004543","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100008277","name":"College of William and Mary","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100008277","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100010909","name":"Young Scientists Fund","doi-asserted-by":"publisher","award":["11701349"],"award-info":[{"award-number":["11701349"]}],"id":[{"id":"10.13039\/501100010909","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004311","name":"Shanxi University","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100004311","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000893","name":"Simons Foundation","doi-asserted-by":"publisher","award":["522728"],"award-info":[{"award-number":["522728"]}],"id":[{"id":"10.13039\/100000893","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Discrete Math."],"published-print":{"date-parts":[[2021,1]]},"abstract":"<jats:p>The 2-dimensional global rigidity has been shown to be equivalent to 3-connectedness and redundant rigidity by a combination of two results due to Jackson and Jord\u00e1n, and Connelly, respectively. By the characterization, a theorem of Lov\u00e1sz and Yemini implies that every 6-connected graph is redundantly rigid and thus globally rigid. The 6-connectedness is best possible, since there exist infinitely many 5-connected nonrigid graphs. Jackson, Servatius, and Servatius used the idea of \u201cessential connectivity\u201d and proved that every 4-connected \u201cessentially 6-connected\u201d graph is redundantly rigid and thus global rigid. Since 3-connectedness is a necessary condition of global rigidity, it is interesting to study 3-connected graphs for redundant rigidity and thus global rigidity. We utilize a different \u201cessential connectivity\u201d and prove that every 3-connected essentially 9-connected graph is redundantly rigid and thus globally rigid. The essential 9-connectedness is best possible. Under this essential connectivity, we also prove that every 4-connected essentially 6-connected graph is redundantly rigid and thus globally rigid. Our proofs are based on discharging arguments.<\/jats:p>","DOI":"10.1137\/20m1375498","type":"journal-article","created":{"date-parts":[[2021,10,19]],"date-time":"2021-10-19T10:50:34Z","timestamp":1634640634000},"page":"2520-2534","source":"Crossref","is-referenced-by-count":1,"title":["Sufficient Conditions for 2-Dimensional Global Rigidity"],"prefix":"10.1137","volume":"35","author":[{"given":"Xiaofeng","family":"Gu","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Wei","family":"Meng","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Martin","family":"Rolek","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yue","family":"Wang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5898-7344","authenticated-orcid":true,"given":"Gexin","family":"Yu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2021,10,19]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1978-0511410-9"},{"key":"atypb2","doi-asserted-by":"crossref","unstructured":"J. A. Bondy and U. S. R. Murty,\n                      Graph Theory\n                      , Springer, New York, 2008.","DOI":"10.1007\/978-1-84628-970-5"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1007\/s00454-004-1124-4"},{"key":"atypb4","doi-asserted-by":"crossref","unstructured":"X. Gu, W. Meng, M. Rolek, Y. Wang, and G. Yu,\n                      Sufficient Conditions for 2-Dimensional Global Rigidity\n                      , preprint,https:\/\/arxiv.org\/abs\/2106.08539, 2021.","DOI":"10.1137\/20M1375498"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1137\/0221008"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2004.11.002"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1016\/j.dam.2008.12.003"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.20196"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(92)90058-6"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2005.11.002"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1007\/BF01534980"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1137\/0603009"},{"key":"atypb13","first-page":"58","volume":"7","author":"Pollaczek-Geiringer H.","year":"1927","journal-title":"Math. Mech."},{"key":"atypb14","first-page":"369","volume":"12","author":"Pollaczek-Geiringer H.","year":"1932","journal-title":"Math. Mech."},{"key":"atypb15","first-page":"171","author":"Whiteley W.","year":"1996","journal-title":"RI"}],"container-title":["SIAM Journal on Discrete Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/20M1375498","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:06:30Z","timestamp":1787317590000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/20M1375498"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1]]},"references-count":15,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2021,1]]}},"alternative-id":["10.1137\/20M1375498"],"URL":"https:\/\/doi.org\/10.1137\/20m1375498","relation":{},"ISSN":["0895-4801","1095-7146"],"issn-type":[{"value":"0895-4801","type":"print"},{"value":"1095-7146","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1]]}}}