A graph is 2-planar if it can be drawn in the plane such that no edge is crossed more than twice.
Bekos et al. (2017) completely characterized optimal non-simple 2- and 3-planar graphs, which have edge counts of 
and 
, respectively.
2-Planar Graph
See also
2-Planar Graph, Doublecross GraphExplore with Wolfram|Alpha
References
Bekos, M. A.; Kaufmann, M.; and Raftopoulou, C. N. "On Optimal 2- and 3-Planar Graphs." In SoCG 2017 (Ed. B. Aronov and M. J. Katz). Vol. 77 of LIPIcs, Schloss Dagstuhl--Leibniz-Zentrum für Informatik, pp. 16:1-16:16, 2017.Brandenburg, F. J. "Straight-Line Drawings of 1-Planar Graphs." 3 Sep 2021. https://arxiv.org/abs/2109.01692.Cite this as:
Weisstein, Eric W. "2-Planar Graph." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/2-PlanarGraph.html