The graph planarization of a graph drawing in the plane with finitely many graph crossings is the planar graph obtained by making each graph crossing a new graph vertex and splitting the edges at those points. In a good graph drawing, each new graph vertex has vertex degree 4.
For 
,
every good graph drawing in the plane
of the complete graph 
has a 3-connected graph
planarization (Pan and Richter 2007). Here, good means that adjacent edges
meet only at their common endpoint, independent edges
meet at most once and only transversely, and no three edges
meet away from vertices.
Is 100." J. Graph Th. 56, 128-134, 2007.