LATIN 2018: LATIN 2018: Theoretical Informatics pp 683-696

# Recognizing Generalized Transmission Graphs of Line Segments and Circular Sectors

• Katharina Klost
• Wolfgang Mulzer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10807)

## 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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