Dense Graphs Have Rigid Parts

Authors Orit E. Raz, József Solymosi

Thumbnail PDF


  • Filesize: 0.52 MB
  • 13 pages

Document Identifiers

Author Details

Orit E. Raz
  • Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Israel
József Solymosi
  • Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada


The authors also thank Omer Angel and Ching Wong for several useful comments regarding the paper.

Cite AsGet BibTex

Orit E. Raz and József Solymosi. Dense Graphs Have Rigid Parts. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 65:1-65:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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 C₀n^{3/2}(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(n^{3/2}log n) is tight, and we provide a construction that shows that requiring Ω(n log n) edges is necessary.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatoric problems
  • Mathematics of computing → Graph theory
  • Graph rigidity
  • line configurations in 3D


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. L. Asimow and B. Roth. The rigidity of graphs. Trans. Amer. Math. Soc., 245:279-289, 1978. Google Scholar
  2. E. D. Bolker and B. Roth. When is a bipartite graph a rigid framework? Pacific J. Math., 90:27-44, 1980. Google Scholar
  3. Gy. Elekes and M. Sharir. Incidences in three dimensions and distinct distances in the plane. Combinat. Probab. Comput., 20:571-608, 2011. Google Scholar
  4. L. Guth and N. H. Katz. On the Erdős distinct distances problem in the plane. Annals Math., 18:155-190, 2015. Google Scholar
  5. J. Kollár. Szemerédi-trotter-type theorems in dimension 3. Adv. Math., 271:30-61, 2015. Google Scholar
  6. G. Laman. On graphs and rigidity of plane skeletal structures. J. Engrg. Math., 4:333-338, 1970. Google Scholar
  7. H. Pollaczek-Geiringer. über die gliederung ebener fachwerke, zamm. Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, 7.1:58-72, 1927. Google Scholar
  8. O. E. Raz. Configurations of lines in space and combinatorial rigidity. Discrete Comput. Geom. (special issue), 58:986-1009, 2017. Google Scholar
  9. O. E. Raz. Distinct distances for points lying on curves in ℝ^d - the bipartite case. manuscript, 2020. Google Scholar