Cubic Planar Graphs that cannot be Drawn on few Lines.

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.