RAC-drawability is $\exists\mathbb{R}$-complete

TL;DR

Deciding if a graph admits a RAC-drawing, where crossings are right-angled, is computationally as hard as solving the existential theory of the reals, even under restricted conditions.

Contribution

The paper proves that the RAC-drawability problem is -hard in the -complete class, establishing its computational complexity.

Findings

RAC-drawability is -complete.

Hardness persists even with limited crossings per edge.

Complexity holds even when drawing is specified up to isomorphism.

Abstract

A RAC-drawing of a graph is a straight-line drawing in which every crossing occurs at a right-angle. We show that deciding whether a graph has a RAC-drawing is as hard as the existential theory of the reals, even if we know that every edge is involved in at most ten crossings and even if the drawing is specified up to isomorphism.