# Dense graphs have rigid parts

## TL;DR

This paper proves that dense enough graphs embedded in the plane must contain a small rigid subframework, using advanced geometric and combinatorial methods, extending previous results on graph rigidity.

## Contribution

It establishes that any sufficiently dense graph with mild general position conditions contains a rigid subframework, connecting graph rigidity with line configurations in three-dimensional space.

## Key findings

- Dense graphs with at least C₀n^{3/2} log n edges contain a rigid subframework.
- A construction shows that fewer than Ω(n log n) edges may not guarantee rigidity.
- The proof extends Guth and Katz's line configuration results with Kollár's additional insights.

## Abstract

While the problem of determining whether an embedding of a graph $G$ in $\mathbb{R}^2$ is {\it infinitesimally rigid} is well understood, specifying whether a given embedding of $G$ is {\it rigid} or not is still a hard task that usually requires ad hoc arguments. In this paper, we show that {\it every} embedding (not necessarily generic) of a dense enough graph (concretely, a graph with at least $C_0n^{3/2}\log n$ edges, for some absolute constant $C_0>0$), 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 [Ra], 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 [GK2]. In fact, our proof requires an extended version of Guth and Katz result; the extension we need is proved by J\'anos Koll\'ar 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.