Let $G$ be simple finite graph of order $n$ and maximum degree $3$.
Can we efficiently draw $G$ with at most $o(n^2)$ crossings?
"Efficiently" means in time polynomial in $n$ or at worst $\exp{o(n^2)}$.
In the drawing the edges are arbitrary curves.
According to a paper
We compute a drawing on the plane of a bounded degree graph in which the sum of the numbers of vertices and crossings is $o(\log^3{n})$ times the optimal minimum sum
so bounding the crossing number will help.
If the crossing number of $G$ is too large the answer is negative.
