Dense graphs have rigid parts

Orit E. Raz, József Solymosi

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While the problem of determining whether an embedding of a graph G in R2 is infinitesimally rigid is well understood, specifying whether a given embedding of G is rigid or not is still a hard task that usually requires ad hoc arguments. In this paper, we show that every embedding (not necessarily generic) of a dense enough graph (concretely, a graph with at least C0n3/2(log n)β edges, for some absolute constants C0 > 0 and β), which satisfies some very mild general position requirements (no three vertices of G are embedded to a common line), must have a subframework of size at least three which is rigid. For the proof we use a connection, established in Raz [Discrete Comput. Geom., 2017], between the notion of graph rigidity and configurations of lines in R3. This connection allows us to use properties of line configurations established in Guth and Katz [Annals Math., 2015]. In fact, our proof requires an extended version of Guth and Katz result; the extension we need is proved by János Kollár in an Appendix to our paper. We do not know whether our assumption on the number of edges being Ω(n3/2 log n) is tight, and we provide a construction that shows that requiring Ω(n log n) edges is necessary.

Original languageAmerican English
Title of host publication36th International Symposium on Computational Geometry, SoCG 2020
EditorsSergio Cabello, Danny Z. Chen
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771436
StatePublished - 1 Jun 2020
Event36th International Symposium on Computational Geometry, SoCG 2020 - Zurich, Switzerland
Duration: 23 Jun 202026 Jun 2020

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
ISSN (Print)1868-8969


Conference36th International Symposium on Computational Geometry, SoCG 2020

Bibliographical note

Publisher Copyright:
© Orit E. Raz and József Solymosi; licensed under Creative Commons License CC-BY 36th International Symposium on Computational Geometry (SoCG 2020).


  • Graph rigidity
  • Line configurations in 3D


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