Geodesic
On plane drawings of $2$-planar graphs
Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part XI, Tome 488 (2019), pp. 49-65

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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). 
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     author = {D. V. Karpov},
     title = {On plane drawings of $2$-planar graphs},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {49--65},
     publisher = {mathdoc},
     volume = {488},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2019_488_a2/}
}
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D. V. Karpov. On plane drawings of $2$-planar graphs. Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part XI, Tome 488 (2019), pp. 49-65. http://geodesic.mathdoc.fr/item/ZNSL_2019_488_a2/