{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T15:34:49Z","timestamp":1759937689770,"version":"3.37.3"},"reference-count":39,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2022,3,19]],"date-time":"2022-03-19T00:00:00Z","timestamp":1647648000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2022,3,19]],"date-time":"2022-03-19T00:00:00Z","timestamp":1647648000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/100010663","name":"H2020 European Research Council","doi-asserted-by":"publisher","award":["675789","675789"],"award-info":[{"award-number":["675789","675789"]}],"id":[{"id":"10.13039\/100010663","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Discrete Comput Geom"],"published-print":{"date-parts":[[2022,10]]},"DOI":"10.1007\/s00454-022-00370-3","type":"journal-article","created":{"date-parts":[[2022,3,19]],"date-time":"2022-03-19T20:02:45Z","timestamp":1647720165000},"page":"796-816","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["New Upper Bounds for the Number of Embeddings of Minimally Rigid Graphs"],"prefix":"10.1007","volume":"68","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3117-9694","authenticated-orcid":false,"given":"Evangelos","family":"Bartzos","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2339-5303","authenticated-orcid":false,"given":"Ioannis Z.","family":"Emiris","sequence":"additional","affiliation":[]},{"given":"Raimundas","family":"Vidunas","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,3,19]]},"reference":[{"key":"370_CR1","doi-asserted-by":"publisher","first-page":"279","DOI":"10.1090\/S0002-9947-1978-0511410-9","volume":"245","author":"L Asimow","year":"1978","unstructured":"Asimow, L., Roth, B.: The rigidity of graphs. Trans. Am. Math. Soc. 245, 279\u2013289 (1978)","journal-title":"Trans. Am. Math. Soc."},{"key":"370_CR2","series-title":"Cambridge Urban and Architectural Studies","volume-title":"Incidence and Symmetry in Design and Architecture","author":"JA Baglivo","year":"1983","unstructured":"Baglivo, J.A., Graver, J.E.: Incidence and Symmetry in Design and Architecture. Cambridge Urban and Architectural Studies, vol. 7. Cambridge University Press, Cambridge (1983)"},{"key":"370_CR3","doi-asserted-by":"publisher","first-page":"189","DOI":"10.1016\/j.jsc.2019.10.015","volume":"102","author":"E Bartzos","year":"2021","unstructured":"Bartzos, E., Emiris, I.Z., Legersk\u00fd, J., Tsigaridas, E.: On the maximal number of real embeddings of minimally rigid graphs in $${\\mathbb{R}}^2$$, $${\\mathbb{R}}^3$$ and $$S^2$$. J. Symbolic Comput. 102, 189\u2013208 (2021)","journal-title":"J. Symbolic Comput."},{"issue":"5\u20136","key":"370_CR4","doi-asserted-by":"publisher","first-page":"325","DOI":"10.1007\/s00200-020-00447-7","volume":"31","author":"E Bartzos","year":"2020","unstructured":"Bartzos, E., Emiris, I.Z., Schicho, J.: On the multihomogeneous B\u00e9zout bound on the number of embeddings of minimally rigid graphs. Appl. Algebra Eng. Commun. Comput. 31(5\u20136), 325\u2013357 (2020)","journal-title":"Appl. Algebra Eng. Commun. Comput."},{"key":"370_CR5","doi-asserted-by":"crossref","unstructured":"Bartzos, E., Emiris, I.Z., Tzamos, Ch.: The m-B\u00e9zout bound and distance geometry. In: Computer Algebra in Scientific Computing\u201423rd Intern. Workshop (Sochi 2021). Lecture Notes in Computer Science, vol. 12865, pp. 6\u201320. Springer, Cham (2021)","DOI":"10.1007\/978-3-030-85165-1_2"},{"key":"370_CR6","doi-asserted-by":"crossref","unstructured":"Berg, A.R., Jord\u00e1n, T.: Algorithms for graph rigidity and scene analysis. In: 11th Annual European Symposium on Algorithms (Budapest 2003). Lecture Notes in Computer Science, vol. 2832, pp. 78\u201389. Springer, Berlin (2003)","DOI":"10.1007\/978-3-540-39658-1_10"},{"issue":"3","key":"370_CR7","doi-asserted-by":"publisher","first-page":"183","DOI":"10.1007\/BF01075595","volume":"9","author":"DN Bernstein","year":"1975","unstructured":"Bernstein, D.N.: The number of roots of a system of equations. Funct. Anal. Appl. 9(3), 183\u2013185 (1975)","journal-title":"Funct. Anal. Appl."},{"issue":"4","key":"370_CR8","doi-asserted-by":"publisher","first-page":"337","DOI":"10.1007\/s10288-016-0314-2","volume":"14","author":"SJL Billinge","year":"2016","unstructured":"Billinge, S.J.L., Duxbury, P.M., Gon\u00e7alves, D.S., Lavor, C., Mucherino, A.: Assigned and unassigned distance geometry: applications to biological molecules and nanostructures. 4OR 14(4), 337\u2013376 (2016)","journal-title":"4OR"},{"key":"370_CR9","volume-title":"Theory and Applications of Distance Geometry","author":"LM Blumenthal","year":"1970","unstructured":"Blumenthal, L.M.: Theory and Applications of Distance Geometry. Chelsea, New York (1970)"},{"issue":"2","key":"370_CR10","doi-asserted-by":"publisher","first-page":"287","DOI":"10.1007\/s00454-003-2902-0","volume":"31","author":"C Borcea","year":"2004","unstructured":"Borcea, C., Streinu, I.: The number of embeddings of minimally rigid graphs. Discrete Comput. Geom. 31(2), 287\u2013303 (2004)","journal-title":"Discrete Comput. Geom."},{"key":"370_CR11","unstructured":"Bregman, L.M.: Some properties of nonnegative matrices and their permanents. Dokl. Akad. Nauk SSSR 211(1), 27\u201330 (1973). (in Russian)"},{"issue":"1","key":"370_CR12","doi-asserted-by":"publisher","first-page":"94","DOI":"10.1137\/17M1118312","volume":"2","author":"J Capco","year":"2018","unstructured":"Capco, J., Gallet, M., Grasegger, G., Koutschan, Ch., Lubbes, N., Schicho, J.: The number of realizations of a Laman graph. SIAM J. Appl. Algebra Geom. 2(1), 94\u2013125 (2018)","journal-title":"SIAM J. Appl. Algebra Geom."},{"key":"370_CR13","doi-asserted-by":"crossref","unstructured":"Emiris, I.Z., Tsigaridas, E.P., Varvitsiotis, A.: Mixed volume and distance geometry techniques for counting Euclidean embeddings of rigid graphs. In: Distance Geometry, pp. 23\u201345. Springer, New York (2013)","DOI":"10.1007\/978-1-4614-5128-0_2"},{"key":"370_CR14","volume-title":"Structures Tendues et Autotendantes","author":"DG Emmerich","year":"1988","unstructured":"Emmerich, D.G.: Structures Tendues et Autotendantes. Editions de La Villette, Paris (1988)"},{"key":"370_CR15","doi-asserted-by":"crossref","unstructured":"Felsner, S., Zickfeld, F.: On the number of planar orientations with prescribed degrees. Electron. J. Combin. 15(1), # 77 (2008)","DOI":"10.37236\/801"},{"key":"370_CR16","doi-asserted-by":"crossref","unstructured":"Gallet, M., Grasegger, G., Schicho, J.: Counting realizations of Laman graphs on the sphere. Electron. J. Combin. 27(2), # 2.5 (2020)","DOI":"10.37236\/8548"},{"issue":"6","key":"370_CR17","doi-asserted-by":"publisher","first-page":"443","DOI":"10.1093\/bfgp\/els023","volume":"11","author":"ME G\u00e1sp\u00e1r","year":"2012","unstructured":"G\u00e1sp\u00e1r, M.E., Csermely, P.: Rigidity and flexibility of biological networks. Brief. Funct. Genom. 11(6), 443\u2013456 (2012)","journal-title":"Brief. Funct. Genom."},{"issue":"2","key":"370_CR18","doi-asserted-by":"publisher","first-page":"125","DOI":"10.1080\/10586458.2018.1437851","volume":"29","author":"G Grasegger","year":"2020","unstructured":"Grasegger, G., Koutschan, Ch., Tsigaridas, E.: Lower bounds on the number of realizations of rigid graphs. Exp. Math. 29(2), 125\u2013136 (2020)","journal-title":"Exp. Math."},{"key":"370_CR19","doi-asserted-by":"publisher","DOI":"10.21236\/AD0705364","volume-title":"Graph Theory","author":"F Harary","year":"1969","unstructured":"Harary, F.: Graph Theory. Addison-Wesley, London (1969)"},{"issue":"1","key":"370_CR20","doi-asserted-by":"publisher","first-page":"71","DOI":"10.1016\/0040-9383(84)90026-0","volume":"23","author":"J Harris","year":"1984","unstructured":"Harris, J., Tu, L.W.: On symmetric and skew-symmetric determinantal varieties. Topology 23(1), 71\u201384 (1984)","journal-title":"Topology"},{"key":"370_CR21","doi-asserted-by":"publisher","first-page":"42","DOI":"10.1016\/j.dam.2017.12.009","volume":"256","author":"B Jackson","year":"2019","unstructured":"Jackson, B., Owen, J.C.: Equivalent realisations of a rigid graph. Discrete Appl. Math. 256, 42\u201358 (2019)","journal-title":"Discrete Appl. Math."},{"key":"370_CR22","unstructured":"Jungnickel, D.: Graphs, Networks and Algorithms. Algorithms and Computation in Mathematics, vol.\u00a05. Springer, Berlin (2005)"},{"issue":"1","key":"370_CR23","doi-asserted-by":"publisher","first-page":"38","DOI":"10.1007\/BF01077562","volume":"12","author":"AG Khovanskii","year":"1978","unstructured":"Khovanskii, A.G.: Newton polyhedra and the genus of complete intersections. Funct. Anal. Appl. 12(1), 38\u201346 (1978)","journal-title":"Funct. Anal. Appl."},{"issue":"1","key":"370_CR24","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/BF01389769","volume":"32","author":"AG Kouchnirenko","year":"1976","unstructured":"Kouchnirenko, A.G.: Poly\u00e8dres de Newton et nombres de Milnor. Invent. Math. 32(1), 1\u201331 (1976)","journal-title":"Invent. Math."},{"issue":"3","key":"370_CR25","doi-asserted-by":"publisher","first-page":"423","DOI":"10.1080\/00207170802108441","volume":"82","author":"L Krick","year":"2009","unstructured":"Krick, L., Broucke, M.E., Francis, B.A.: Stabilisation of infinitesimally rigid formations of multi-robot networks. Int. J. Control 82(3), 423\u2013439 (2009)","journal-title":"Int. J. Control"},{"issue":"4","key":"370_CR26","doi-asserted-by":"publisher","first-page":"331","DOI":"10.1007\/BF01534980","volume":"4","author":"G Laman","year":"1970","unstructured":"Laman, G.: On graphs and rigidity of plane skeletal structures. J. Eng. Math. 4(4), 331\u2013340 (1970)","journal-title":"J. Eng. Math."},{"issue":"182","key":"370_CR27","doi-asserted-by":"publisher","first-page":"294","DOI":"10.1080\/14786446408643668","volume":"27","author":"JC Maxwell","year":"1864","unstructured":"Maxwell, J.C.: On the calculation of the equilibrium and stiffness of frames. Philos. Mag. 27(182), 294\u2013299 (1864)","journal-title":"Philos. Mag."},{"key":"370_CR28","doi-asserted-by":"publisher","first-page":"789","DOI":"10.1090\/S0002-9904-1963-11031-9","volume":"69","author":"H Minc","year":"1963","unstructured":"Minc, H.: Upper bounds for permanents of $$(0,1)$$-matrices. Bull. Am. Math. Soc. 69, 789\u2013791 (1963)","journal-title":"Bull. Am. Math. Soc."},{"issue":"1","key":"370_CR29","doi-asserted-by":"publisher","first-page":"58","DOI":"10.1002\/zamm.19270070107","volume":"7","author":"H Pollaczek-Geiringer","year":"1927","unstructured":"Pollaczek-Geiringer, H.: \u00dcber die Gliederung ebener Fachwerke. Zeitschrift f\u00fcr Angewandte Mathematik und Mechanik 7(1), 58\u201372 (1927)","journal-title":"Zeitschrift f\u00fcr Angewandte Mathematik und Mechanik"},{"issue":"6","key":"370_CR30","doi-asserted-by":"publisher","first-page":"369","DOI":"10.1002\/zamm.19320120606","volume":"12","author":"H Pollaczek-Geiringer","year":"1932","unstructured":"Pollaczek-Geiringer, H.: Zur Gliederungstheorie r\u00e4umlicher Fachwerke. Zeitschrift f\u00fcr Angewandte Mathematik und Mechanik 12(6), 369\u2013376 (1932)","journal-title":"Zeitschrift f\u00fcr Angewandte Mathematik und Mechanik"},{"key":"370_CR31","unstructured":"Schulze, B., Whiteley, W.: Rigidity and scene analysis. In: Handbook of Discrete and Computational Geometry. CRC Press Ser. Discrete Math. Appl., pp. 1593\u20131632. CRC Press, Boca Raton (1997)"},{"key":"370_CR32","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-38010-5","volume-title":"Basic Algebraic Geometry 1. Varieties in Projective Space","author":"IR Shafarevich","year":"2013","unstructured":"Shafarevich, I.R.: Basic Algebraic Geometry 1. Varieties in Projective Space. Springer, Heidelberg (2013)"},{"issue":"18","key":"370_CR33","doi-asserted-by":"publisher","first-page":"3028","DOI":"10.1016\/j.dam.2013.06.004","volume":"161","author":"O Shai","year":"2013","unstructured":"Shai, O., Sljoka, A., Whiteley, W.: Directed graphs, decompositions, and spatial linkages. Discrete Appl. Math. 161(18), 3028\u20133047 (2013)","journal-title":"Discrete Appl. Math."},{"issue":"2","key":"370_CR34","doi-asserted-by":"publisher","first-page":"84","DOI":"10.1016\/j.comgeo.2009.04.004","volume":"43","author":"R Steffens","year":"2010","unstructured":"Steffens, R., Theobald, T.: Mixed volume techniques for embeddings of Laman graphs. Comput. Geom. 43(2), 84\u201393 (2010)","journal-title":"Comput. Geom."},{"issue":"8","key":"370_CR35","doi-asserted-by":"publisher","first-page":"1944","DOI":"10.1016\/j.ejc.2008.12.018","volume":"30","author":"I Streinu","year":"2009","unstructured":"Streinu, I., Theran, L.: Sparse hypergraphs and pebble game algorithms. Eur. J. Combin. 30(8), 1944\u20131964 (2009)","journal-title":"Eur. J. Combin."},{"key":"370_CR36","first-page":"21","volume":"11","author":"T-S Tay","year":"1985","unstructured":"Tay, T.-S., Whiteley, W.: Generating isostatic frameworks. Struct. Topol. 11, 21\u201369 (1985)","journal-title":"Struct. Topol."},{"key":"370_CR37","first-page":"53","volume":"8","author":"W Whiteley","year":"1983","unstructured":"Whiteley, W.: Cones, infinity and $$1$$-story buildings. Struct. Topol. 8, 53\u201370 (1983)","journal-title":"Struct. Topol."},{"key":"370_CR38","doi-asserted-by":"crossref","unstructured":"Zelazo, D., Franchi, A., Allg\u00f6wer, F., B\u00fclthoff, H.H., Giordano, P.R.: Rigidity maintenance control for multi-robot systems. In: Robotics: Science and Systems VIII (Sydney 2012), # 60. MIT Press, Cambridge (2013)","DOI":"10.15607\/RSS.2012.VIII.060"},{"issue":"6","key":"370_CR39","doi-asserted-by":"publisher","first-page":"3059","DOI":"10.1137\/090772009","volume":"20","author":"Z Zhu","year":"2010","unstructured":"Zhu, Z., So, A.M.-C., Ye, Y.: Universal rigidity and edge sparsification for sensor network localization. SIAM J. Optim. 20(6), 3059\u20133081 (2010)","journal-title":"SIAM J. Optim."}],"container-title":["Discrete & Computational Geometry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-022-00370-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00454-022-00370-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-022-00370-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,9,12]],"date-time":"2022-09-12T19:03:40Z","timestamp":1663009420000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00454-022-00370-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,3,19]]},"references-count":39,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2022,10]]}},"alternative-id":["370"],"URL":"https:\/\/doi.org\/10.1007\/s00454-022-00370-3","relation":{},"ISSN":["0179-5376","1432-0444"],"issn-type":[{"type":"print","value":"0179-5376"},{"type":"electronic","value":"1432-0444"}],"subject":[],"published":{"date-parts":[[2022,3,19]]},"assertion":[{"value":"16 October 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 April 2021","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 October 2021","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 March 2022","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}