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Contact Representations of Graphs in 3D

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Algorithms and Data Structures (WADS 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9214))

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Abstract

We study contact representations of non-planar graphs in which vertices are represented by axis-aligned polyhedra in 3D and edges are realized by non-zero area common boundaries between corresponding polyhedra. We present a liner-time algorithm constructing a representation of a 3-connected planar graph, its dual, and the vertex-face incidence graph with 3D boxes. We then investigate contact representations of 1-planar graphs. We first prove that optimal 1-planar graphs without separating 4-cycles admit a contact representation with 3D boxes. However, since not every optimal 1-planar graph can be represented in this way, we also consider contact representations with the next simplest axis-aligned 3D object, L-shaped polyhedra. We provide a quadratic-time algorithm for representing optimal 1-planar graphs with L-shapes.

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Authors and Affiliations

  1. Department of Computer Science, University of Arizona, Tucson, USA

    Jawaherul Alam, Stephen Kobourov, Sergey Pupyrev & Jackson Toeniskoetter

  2. Department of Computer Science, University of British Columbia, Vancouver, Canada

    William Evans

  3. Institute of Mathematics and Computer Science, Ural Federal University, Yekaterinburg, Russia

    Sergey Pupyrev

  4. Department of Mathematics, Karlsruhe Institute of Technology, Karlsruhe, Germany

    Torsten Ueckerdt

Authors
  1. Jawaherul Alam
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  2. William Evans
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  3. Stephen Kobourov
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  4. Sergey Pupyrev
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  5. Jackson Toeniskoetter
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  6. Torsten Ueckerdt
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Corresponding author

Correspondence to Jawaherul Alam .

Editor information

Editors and Affiliations

  1. Carleton University, Ottawa, Canada

    Frank Dehne

  2. Carleton University, Ottawa, Canada

    Jörg-Rüdiger Sack

  3. University of Victoria, Victoria, Canada

    Ulrike Stege

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Alam, J., Evans, W., Kobourov, S., Pupyrev, S., Toeniskoetter, J., Ueckerdt, T. (2015). Contact Representations of Graphs in 3D. In: Dehne, F., Sack, JR., Stege, U. (eds) Algorithms and Data Structures. WADS 2015. Lecture Notes in Computer Science(), vol 9214. Springer, Cham. https://doi.org/10.1007/978-3-319-21840-3_2

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  • DOI: https://doi.org/10.1007/978-3-319-21840-3_2

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-21839-7

  • Online ISBN: 978-3-319-21840-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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