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.

翻译：我们称一个（多）图$G = (V,E)$具有几何厚度$t$，若存在一个直线绘制$\varphi : V \rightarrow \mathbb{R}^2$以及对其边的一种$t$-着色，使得在其相对内部共享点的任意两条边不具有相同颜色。\textsc{几何厚度}问题询问给定多图是否具有至多$t$的几何厚度。该问题已被证明在$t=2$时是NP-难的[Durocher, Gethner, and Mondal, CG 2016]。本文中，我们通过证明该问题在厚度为$30$时已是$\exists \mathbb{R}$-完全的，从而确定了\textsc{几何厚度}的计算复杂性。此外，我们的归约表明该问题对于$4392$-平面图是$\exists \mathbb{R}$-完全的，其中若一个图允许每条边至多$k$次交叉的拓扑绘制，则称其为$k$-平面图。在研究过程中，我们解答了先前关于几何厚度及其他相关问题的疑问，特别是证明了31个边不相交图的同步图嵌入问题以及色数为30的伪线段可拉伸性问题是$\exists \mathbb{R}$-完全的。