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arXiv · 2301.09246

On the Biplanarity of Blowups

David Eppstein
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Abstract

The 2-blowup of a graph is obtained by replacing each vertex with two non-adjacent copies; a graph is biplanar if it is the union of two planar graphs. We disprove a conjecture of Gethner that 2-blowups of planar graphs are biplanar: iterated Kleetopes are counterexamples. Additionally, we construct biplanar drawings of 2-blowups of planar graphs whose duals have two-path induced path partitions, and drawings with split thickness two of 2-blowups of 3-chromatic planar graphs, and of graphs that can be decomposed into a Hamiltonian path and a dual Hamiltonian path.

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David Eppstein. 2023-09-01. On the Biplanarity of Blowups. https://doi.org/10.7155/jgaa.v28i2.2989

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