Can any non-planar graph with n minimum crossing points be 'drawn' on a sphere so the vertice and edge sets are the same and it has a connected subset A with minimum r crossing points and a disjoint connected subset B with minimum s crossing points where r+s=n? And other subsets can be found for any value r such that 1 < r < n ? 



About planar graphs?
No. Suppose it could, then you could just apply the steriographic projection from an interior point of some face on the sphere and obtain a graph on the plane with no crossing edges that is isomorphic to yours. Hence your graph must have been planar.


