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A graph is 1-planar if it can be drawn on the plane so that each edge iscrossed by no more than one other edge (and any pair of crossing edges crossonly once). A non-1-planar graph $G$ is minimal if the graph $G-e$ is 1-planarfor every edge $e$ of $G$. We construct two infinite families of minimalnon-1-planar graphs and show that for every integer $n > 62$, there are atleast $2^{(n-54)/4}$ nonisomorphic minimal non-1-planar graphs of order $n$. Itis also proved that testing 1-planarity is NP-complete.

Minimal obstructions for 1-immersions and hardness of 1-planarity testing