On Representing Graphs by Touching Cuboids

  • David Bremner
  • William Evans
  • Fabrizio Frati
  • Laurie Heyer
  • Stephen G. Kobourov
  • William J. Lenhart
  • Giuseppe Liotta
  • David Rappaport
  • Sue H. Whitesides
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7704)


We consider contact representations of graphs where vertices are represented by cuboids, i.e. interior-disjoint axis-aligned boxes in 3D space. Edges are represented by a proper contact between the cuboids representing their endvertices. Two cuboids make a proper contact if they intersect and their intersection is a non-zero area rectangle contained in the boundary of both. We study representations where all cuboids are unit cubes, where they are cubes of different sizes, and where they are axis-aligned 3D boxes. We prove that it is NP-complete to decide whether a graph admits a proper contact representation by unit cubes. We also describe algorithms that compute proper contact representations of varying size cubes for relevant graph families. Finally, we give two new simple proofs of a theorem by Thomassen stating that all planar graphs have a proper contact representation by touching cuboids.


Planar Graph Unit Cube Edge Contact Face Contact Contact Graph 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • David Bremner
    • 1
  • William Evans
    • 2
  • Fabrizio Frati
    • 3
  • Laurie Heyer
    • 4
  • Stephen G. Kobourov
    • 5
  • William J. Lenhart
    • 6
  • Giuseppe Liotta
    • 7
  • David Rappaport
    • 8
  • Sue H. Whitesides
    • 9
  1. 1.Faculty of Computer ScienceUniversity of New BrunswickCanada
  2. 2.Department of Computer ScienceUniversity of British ColumbiaCanada
  3. 3.School of Information TechnologiesThe University of SydneyAustralia
  4. 4.Department of MathematicsDavidson CollegeUSA
  5. 5.Department of Computer ScienceUniversity of ArizonaUSA
  6. 6.Department of Computer ScienceWilliams CollegeUSA
  7. 7.Department of Computer ScienceUniversity of PerugiaItaly
  8. 8.School of ComputingQueens UniversityCanada
  9. 9.Department of Computer ScienceUniversity of VictoriaCanada

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