Contact Representations of Graphs in 3D

  • Jawaherul Alam
  • William Evans
  • Stephen Kobourov
  • Sergey Pupyrev
  • Jackson Toeniskoetter
  • Torsten Ueckerdt
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9214)


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.


Dual Graph Outer Face Primal Graph Contact Representation Schnyder Wood 
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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Jawaherul Alam
    • 1
  • William Evans
    • 2
  • Stephen Kobourov
    • 1
  • Sergey Pupyrev
    • 1
    • 3
  • Jackson Toeniskoetter
    • 1
  • Torsten Ueckerdt
    • 4
  1. 1.Department of Computer ScienceUniversity of ArizonaTucsonUSA
  2. 2.Department of Computer ScienceUniversity of British ColumbiaVancouverCanada
  3. 3.Institute of Mathematics and Computer ScienceUral Federal UniversityYekaterinburgRussia
  4. 4.Department of MathematicsKarlsruhe Institute of TechnologyKarlsruheGermany

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