Dense Graphs Have Rigid Parts
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.
Graph rigidity
line configurations in 3D
Mathematics of computing~Combinatoric problems
Mathematics of computing~Graph theory
65:1-65:13
Regular Paper
The authors also thank Omer Angel and Ching Wong for several useful comments regarding the paper.
Orit E.
Raz
Orit E. Raz
Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Israel
József
Solymosi
József Solymosi
Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada
The work of the second author has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 741420, 617747, 648017). His research is also supported by NSERC and OTKA (K 119528) grants.
10.4230/LIPIcs.SoCG.2020.65
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Orit E. Raz and József Solymosi
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