eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-06-08
65:1
65:13
10.4230/LIPIcs.SoCG.2020.65
article
Dense Graphs Have Rigid Parts
Raz, Orit E.
1
Solymosi, József
2
Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Israel
Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol164-socg2020/LIPIcs.SoCG.2020.65/LIPIcs.SoCG.2020.65.pdf
Graph rigidity
line configurations in 3D