A graph is $k$-planar if it can be drawn in the plane so that each edge is crossed at most $k$ times. A $k$-planar graph $G$ is maximal if $G+uv$ is not $k$-planar for any non-adjacent vertices $u,v\in V(G)$.
- Is there any maximal $1$-planar graph that is not $3$-connected ?
- Is there any maximal $2$-planar graph that is not $3$-connected ?
I believe those examples exist but didn't find any references mentioned this.