The Complexity of Angular Resolution - Schaefer, Marcus (Journal of Graph Algorithms and Applications, 2023)
The angular resolution of a straight-line drawing of a graph is the smallest angle formed by any two edges incident to a vertex. The angular resolution of a graph is the supremum of the angular resolutions of all straight-line drawings of the graph. We show that testing whether a graph has angular resolution at least π/(2k) is complete for R, the existential theory of the reals, for every fixed k ≥ 2. This remains true if the graph is planar and a plane embedding of the graph is fixed.