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Cubic Planar Graphs that cannot be Drawn on few Lines

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arxiv 1903.05256 v1 pith:YGWHYAM6 submitted 2019-03-12 cs.CG

classification cs.CG
keywords graphslinesplanarcannotcubicdrawnbipartitevertices
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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.

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  1. Line and Plane Cover Numbers Revisited

    cs.CG 2019-08 conditional novelty 8.0 of 10

    It is NP-hard to decide whether a planar graph can be drawn with all vertices on two straight lines, and any graph drawable on two planes has at most 5n minus 19 edges.

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