On plane drawings of $2$-planar graphs

It is proved that any $(2k+1)$-edge connected $k$-planar graph has a plane drawing such that any two crossing edges in this drawing cross each other exactly once. It is proved that any $2$-planar graph has a plane drawing such that any two crossing edges in this drawing has no common end and cross each other exactly once. It is also proved that any $2$-planar graph has a supergraph on the same vertex set which can be drawn such that, for any vertex $v$, among every three successive edges incident to $v$, there is at least one simple edge. (An edge is called simple if it does not intersect any other edge in this drawing).