  For every integer $\ell$, we construct a cubic 3-vertex-connected planar
bipartite graph $G$ with $O(\ell^3)$ vertices such that there is no planar
straight-line drawing of $G$ whose vertices all lie on $\ell$ lines. This
strengthens previous results on graphs that cannot be drawn on few lines, which
constructed significantly larger maximal planar graphs. We also find apex-trees
and cubic bipartite series-parallel graphs that cannot be drawn on a bounded
number of lines.
