Cubic Planar Graphs that cannot be Drawn on few Lines
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For every integer ℓ, we construct a cubic 3-vertex-connected planar bipartite graph G with O(ℓ^3) vertices such that there is no planar straight-line drawing of G whose vertices all lie on ℓ 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.
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