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

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