Complexity of Geometric k-Planarity for Fixed k

Abstract

The rectilinear local crossing number, $\mathop{\overline{\rm lcr}}(G)$, of a graph $G$ is the smallest $k$ so that $G$ has a straight-line drawing with at most $k$ crossings along each edge. We show that deciding whether $\mathop{\overline{\rm lcr}}(G) \leq k$ for a fixed $k$ is complete for the existential theory of the reals, $\exists \mathbb{R}$.