Dense Graphs Have Rigid Parts

Abstract

While the problem of determining whether an embedding of a graph G in $${\mathbb {R}}^2$$ 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 $$C_0n^{3/2}(\log n)^{\beta }$$ edges, for some absolute constants $$C_0>0$$ and $$\beta $$ ), 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. 58(4), 986–1009 (2017)), between the notion of graph rigidity and configurations of lines in $${\mathbb {R}}^3$$ . This connection allows us to use properties of line configurations established in Guth and Katz (Ann. Math. 181(1), 155–190 (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 $$\Omega (n^{3/2}\log n)$$ is tight, and we provide a construction that shows that requiring $$\Omega (n\log n)$$ edges is necessary.