A planar graph is such that one can draw it on the plane so that edges do not intersect except at vertices. Consider a weaker condition:


* We can draw the graph on a plane so that for every two edges that intersect properly (that is, not at a vertex
of the graph) there are no edges that intersect properly both these
edges.


What is known about such graphs? 


Edit: What if we strengthen the condition (similar to [outplanarity](http://mathworld.wolfram.com/OutplanarGraph.html)): 


* We can draw the graph on a plane so that all vertices belong to a circle and for every two edges that intersect properly (that is, not at a vertex
of the graph) there are no edges that intersect properly both these
edges.



A weak version of planarity
A graph drawn in the plane (with edges represented by curves that do not pass through vertices except for their endpoints) is called *$k$-quasi-planar* if there are is no set of $k$ pairwise intersecting curves. For $k=3$ they are usually just called *quasi-planar*.


Building on earlier work by Agarwal et al (doi:10.1007/BF01196127), Pach, Radoicic and Toth showed in "Relaxing planarity for topological graphs" that there is a constant $C$ such that every quasi-planar graph on $n$ vertices has at most $Cn$ edges (Theorem 7.1).


