# Is there any maximal 1-planar or 2-planar graph that is not 3-connected

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.

This is a non-3-connected 1-planar example...

The paper On Properties of Maximal 1-planar Graphs by Dávid Hudák, Tomáš Madaras and Yusuke Suzuki describes the construction of maximal 1-planar graphs with minimum degree 2 and with (relatively) small number of edges, namely, (8/3)*(n-2) (which is far less than the number of edges of a maximal n-vertex planar graph).