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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 onlyif $p_j \in x_i$. Generalized transmission graphs provide a generalized modelof the connectivity in networks of directional antennas.  The complexity class $\exists \mathbb{R}$ contains all problems that can bereduced 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 ofthe existential theory of the reals. It lies between $\mathbf{NP}$ and$\mathbf{PSPACE}$.  Many geometric decision problems, such as recognition of disk graphs and ofintersection graphs of lines, are complete for $\exists \mathbb{R}$. Continuingthis line of research, we show that the recognition problem of generalizedtransmission graphs of line segments and of circular sectors is hard for$\exists \mathbb{R}$. As far as we know, this constitutes the first such resultfor a class of directed graphs.

Recognizing Generalized Transmission Graphs of Line Segments and Circular Sectors