# Cubic Planar Graphs that cannot be Drawn on few Lines

## TL;DR

This paper constructs small cubic planar graphs that cannot be drawn with all vertices on a limited number of lines, advancing understanding of geometric constraints in graph drawings.

## Contribution

It introduces the first small cubic 3-connected planar bipartite graphs that cannot be drawn on few lines, improving prior results with larger graphs.

## Key findings

- Constructed cubic 3-connected planar bipartite graphs with O(ℓ^3) vertices that defy few-line drawings.
- Extended results to apex-trees and cubic bipartite series-parallel graphs.
- Demonstrated limitations of line-based drawings for specific graph classes.

## Abstract

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.