WADS 2015: Algorithms and Data Structures pp 14-27

# 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)

## 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.

## Keywords

Dual Graph Outer Face Primal Graph Contact Representation Schnyder Wood
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© 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