On the Biplanarity of Blowups

TL;DR

This paper disproves a conjecture that 2-blowups of planar graphs are biplanar, providing counterexamples and new constructions of biplanar drawings for specific classes of graphs.

Contribution

It disproves Gethner's conjecture by constructing counterexamples using iterated Kleetopes and introduces new methods for biplanar drawings of 2-blowups.

Findings

Disproved Gethner's conjecture on 2-blowups of planar graphs.

Constructed biplanar drawings for 2-blowups with specific dual properties.

Provided drawings with split thickness two for certain complex graphs.

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.