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 Geometric Thickness problem asks whether a given multigraph has geometric thickness at most t. In this paper, we settle the computational complexity of Geometric Thickness by showing that it is $\exists \mathbb {R}$-complete already for thickness $57$. Moreover, our reduction shows that the problem is $\exists \mathbb {R}$-complete for $8280$-planar graphs, where a graph is k-planar if it admits a topological drawing with at most k crossings per edge. In this paper we answer previous questions on geometric thickness and on other related problems, in particular that simultaneous graph embeddings of $58$ edge-disjoint graphs and pseudo-segment stretchability with chromatic number $57$ are $\exists \mathbb {R}$-complete.