Geometric Thickness of Multigraphs is $\exists \mathbb{R}$-complete

Abstract

We say that a (multi)graph $G = (V,E)$ has geometric thickness $t$ if there exists a straight-line drawing $\varphi : V \rightarrow \mathbb{R}^2$ and a $t$-coloring of its edges where no two edges sharing a point in their relative interior have the same color. The \textsc{Geometric Thickness} problem asks whether a given multigraph has geometric thickness at most $t$. This problem was shown to be NP-hard for $t=2$ [Durocher, Gethner, and Mondal, CG 2016]. In this paper, we settle the computational complexity of \textsc{Geometric Thickness} by showing that it is $\exists \mathbb{R}$-complete already for thickness $30$. Moreover, our reduction shows that the problem is $\exists \mathbb{R}$-complete for $4392$-planar graphs, where a graph is $k$-planar if it admits a topological drawing with at most $k$ crossings per edge. In the course of our paper, we answer previous questions on geometric thickness and on other related problems, in particular that simultaneous graph embeddings of $31$ edge-disjoint graphs and pseudo-segment stretchability with chromatic number $30$ are $\exists \mathbb{R}$-complete.