  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.
