On Representing Graphs by Touching Cuboids

Bremner, David and Evans, William S. and Frati, Fabrizio and Heyer, Laurie and Kobourov, Stephen G. and Lenhart, William J. and Liotta, Giuseppe and Rappaport, David and Whitesides, Sue (2013) On Representing Graphs by Touching Cuboids. In: 20th International Symposium, GD 2012, September 19-21, 2012, Redmond, WA, USA , pp. 187-198 (Official URL: http://link.springer.com/chapter/10.1007/978-3-642-36763-2_17).

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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.

Item Type:Conference Paper
Additional Information:10.1007/978-3-642-36763-2_17
Classifications:P Styles > P.060 3D
Z Theory > Z.500 Representations
ID Code:1309

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