Geometric Thickness of Simple Graphs

Determine the computational complexity of deciding the geometric thickness of simple graphs.

Geometric thickness asks whether edges can be partitioned into a fixed number of crossing-free straight-line layers.

The multigraph variant is ER-complete for fixed parameters, but the simple graph case remains unresolved.

The complexity of the geometric thickness of (simple) graphs~\ourref[Graph(s)!Geometric Thickness (Open)]{p:geothick} remains open.

— The Existential Theory of the Reals as a Complexity Class: A Compendium (2407.18006 - Schaefer et al., 25 Jul 2024) in ProblemCG “Geometric Thickness of Multigraph” entry