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On the Biplanarity of Blowups

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Graph Drawing and Network Visualization (GD 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14465))

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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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Notes

  1. 1.

    Blowups are a standard concept but their notation varies significantly. Other choices from the literature include G(k), G[k], \(G^k\), and \(G^{(k)}\).

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Acknowledgements

This research was supported in part by NSF grant CCF-2212129.

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Authors and Affiliations

  1. Computer Science Department, University of California, Irvine, Irvine, USA

    David Eppstein

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  1. David Eppstein
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Correspondence to David Eppstein .

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Editors and Affiliations

  1. University Ioannina, Ioannina, Greece

    Michael A. Bekos

  2. University of Ioannina, Osnabrück, Germany

    Markus Chimani

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Eppstein, D. (2023). On the Biplanarity of Blowups. In: Bekos, M.A., Chimani, M. (eds) Graph Drawing and Network Visualization. GD 2023. Lecture Notes in Computer Science, vol 14465. Springer, Cham. https://doi.org/10.1007/978-3-031-49272-3_1

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  • DOI: https://doi.org/10.1007/978-3-031-49272-3_1

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  • Print ISBN: 978-3-031-49271-6

  • Online ISBN: 978-3-031-49272-3

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