Configurations of lines in space and combinatorial rigidity

Let $L$ be a sequence $(\ell_1,\ell_2,\ldots,\ell_n)$ of $n$ lines in $\mathbb{C}^3$. We define the {\it intersection graph} $G_L=([n],E)$ of $L$, where $[n]:=\{1,\ldots, n\}$, and with $\{i,j\}\in E$ if and only if $i\neq j$ and the corresponding lines $\ell_i$ and $\ell_j$ intersect, or are parallel (or coincide). For a graph $G=([n],E)$, we say that a sequence $L$ is a {\it realization} of $G$ if $G\subset G_L$. One of the main results of this paper is to provide a combinatorial characterization of graphs $G=([n],E)$ that have the following property: For every {\it generic} realization $L$ of $G$ that consists of $n$ pairwise distinct lines, we have $G_L=K_n$, in which case the lines of $L$ are either all concurrent or all coplanar. The general statements that we obtain about lines, apart from their independent interest, turns out to be closely related to the notion of graph rigidity. The connection is established due to the so-called Elekes--Sharir framework, which allows us to transform the problem into an incidence problem involving lines in three dimensions. By exploiting the geometry of contacts between lines in 3D, we can obtain alternative, simpler, and more precise characterizations of the rigidity of graphs.