Recognizing Generalized Transmission Graphs of Line Segments and Circular Sectors

TL;DR

This paper proves that recognizing generalized transmission graphs of line segments and circular sectors is computationally hard for the existential theory of the reals, extending complexity results to directed geometric graphs.

Contribution

It establishes the first complexity hardness results for recognizing directed geometric graphs within the existential theory of the reals class.

Findings

Recognition of these graphs is      for directed geometric graphs.

The problem is  for line segments and circular sectors.

This extends the understanding of complexity in geometric graph recognition to directed graphs.

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.