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On the Planar Split Thickness of Graphs

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Abstract

Motivated by applications in graph drawing and information visualization, we examine the planar split thickness of a graph, that is, the smallest k such that the graph is k-splittable into a planar graph. A k-split operation substitutes a vertex v by at most k new vertices such that each neighbor of v is connected to at least one of the new vertices. We first examine the planar split thickness of complete graphs, complete bipartite graphs, multipartite graphs, bounded degree graphs, and genus-1 graphs. We then prove that it is NP-hard to recognize graphs that are 2-splittable into a planar graph, and show that one can approximate the planar split thickness of a graph within a constant factor. If the treewidth is bounded, then we can even verify k-splittability in linear time, for a constant k.

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Notes

  1. A graph G is k-degenerate if every subgraph of G contains a vertex of degree at most k.

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Acknowledgements

Most of the results of this paper were obtained at the McGill-INRIA-UVictoria Workshop on Computational Geometry, Barbados, February 2015. We would like to thank the organizers of these events, as well as many participants for fruitful discussions and suggestions. The first, fourth, sixth and eighth authors acknowledge the support from NSF Grant 1228639, 2012C4E3KT PRIN Italian National Research Project, PEPS egalite project and NSERC, respectively.

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Authors and Affiliations

  1. University of California, Irvine, CA, USA

    David Eppstein

  2. FernUniversität Hagen, Hagen, Germany

    Philipp Kindermann

  3. University of Arizona, Tucson, AZ, USA

    Stephen Kobourov

  4. Università degli Studi di Perugia, Perugia, Italy

    Giuseppe Liotta

  5. University of Waterloo, Waterloo, Canada

    Anna Lubiw, Debajyoti Mondal & Hamideh Vosoughpour

  6. Universit. Grenoble Alpes, Grenoble, France

    Aude Maignan

  7. University of Victoria, Victoria, Canada

    Sue Whitesides

  8. University of Lethbridge, Lethbridge, Canada

    Stephen Wismath

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  1. David Eppstein
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  2. Philipp Kindermann
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  3. Stephen Kobourov
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  4. Giuseppe Liotta
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  5. Anna Lubiw
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  6. Aude Maignan
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  7. Debajyoti Mondal
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  8. Hamideh Vosoughpour
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  9. Sue Whitesides
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  10. Stephen Wismath
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Corresponding author

Correspondence to Philipp Kindermann.

Additional information

A preliminary version of this work appeared at the 12th Latin American Theoretical Informatics Symposium (LATIN’16) [10].

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Eppstein, D., Kindermann, P., Kobourov, S. et al. On the Planar Split Thickness of Graphs. Algorithmica 80, 977–994 (2018). https://doi.org/10.1007/s00453-017-0328-y

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