While the problem of determining whether an embedding of a graph *G* in *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 the talk I will discuss a recent result (joint with Jozsef Solymosi), where we show that every embedding of a sufficiently dense graph has a rigid subset. The proof uses a reduction of the original rigidity problem to a question about line configurations in *R^3*.