Construct graph such that neither $G$ nor $\overline{G}$ is planar

Question

How can I construct a graph of order $8$ (not necessarily connected) such that neither $G$ nor its complement graph is planar?

From researching myself, I guess that it might have something to do with containing a $K_{3, 3}$ graph? I am not completely sure though and I would like some reassurance on how to solve the problem. I am trying to get better in graph theory. Thanks

Answer 1

I suppose a simple answer is $K_{5,3}$. It contains $K_{3,3}$ and its complement contains $K_5$.

Answer 2

Here's another simple example. A $K_{3,3}$ is nonplanar, and has nine edges, so its complement has $19$ edges (since $K_8$ has $8\times7\div2=28$ edges), and $19>3\times8-6=18$, but any planar graph on $v$ vertices can have at most $3v-6$ edges.