Crossing Numbers of Beyond-Planar Graphs.

We study the 1-planar, quasi-planar, and fan-planar crossing number in comparison to the (unrestricted) crossing number of graphs. We prove that there are n-vertex 1-planar (quasi-planar, fan-planar) graphs such that any 1-planar (quasi-planar, fan-planar) drawing has \(\varOmega (n)\) crossings, while \(\mathcal {O}(1)\) crossings suffice in a crossing-minimal drawing without restrictions on local edge crossing patterns.