Geometric Thickness of Multigraphs is $\exists \mathbb{R}$-complete
Latin American Symposium on Theoretical Informatics(2023)
摘要
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 $t$. 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 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 the course of our paper, we
answer previous questions on the 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.
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