  Suppose we have an arrangement $\mathcal{A}$ of $n$ geometric objects $x_1,
\dots, x_n \subseteq \mathbb{R}^2$ in the plane, with a distinguished point
$p_i$ in each object $x_i$. The generalized transmission graph of $\mathcal{A}$
has vertex set $\{x_1, \dots, x_n\}$ and a directed edge $x_ix_j$ if and only
if $p_j \in x_i$. Generalized transmission graphs provide a generalized model
of the connectivity in networks of directional antennas.
  The complexity class $\exists \mathbb{R}$ contains all problems that can be
reduced in polynomial time to an existential sentence of the form $\exists x_1,
\dots, x_n: \phi(x_1,\dots, x_n)$, where $x_1,\dots, x_n$ range over
$\mathbb{R}$ and $\phi$ is a propositional formula with signature $(+, -,
\cdot, 0, 1)$. The class $\exists \mathbb{R}$ aims to capture the complexity of
the existential theory of the reals. It lies between $\mathbf{NP}$ and
$\mathbf{PSPACE}$.
  Many geometric decision problems, such as recognition of disk graphs and of
intersection graphs of lines, are complete for $\exists \mathbb{R}$. Continuing
this line of research, we show that the recognition problem of generalized
transmission graphs of line segments and of circular sectors is hard for
$\exists \mathbb{R}$. As far as we know, this constitutes the first such result
for a class of directed graphs.
