Let $G$ simple connected graph on $n$ vertices and assume that both $G$ and $G'$(complement) are planar.
$m$ and $m'$ be the number of edges in $G$ and $G$.
$m+m'$ $=$ $n(n-1)/2$
$m, m'$ $≤ 3n − 6$
$n(n−1)/2 =m+m' ≤6n−12$
$⇒$ $n^2 −13n+24≤0$ $⇒$ $n<11$.
Would this be a correct solution?
I have also noticed this only works for connected graphs so I was wondering how would I expand it to disconnected graphs?
Any help would be really appreciated.