# Recognizing Generalized Transmission Graphs of Line Segments and Circular Sectors

## Abstract

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.

## Notes

### Acknowledgments

We would like to thank an anonymous reviewer for pointing out a mistake in Observation 4.1.

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