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):

- 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.

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