On representing graphs by touching cuboids

Authors:
David Bremner;William Evans;Fabrizio Frati;Laurie Heyer;Stephen G. Kobourov;William J. Lenhart;Giuseppe Liotta;David Rappaport;Sue H. Whitesides
Affiliations:
Faculty of Computer Science, University of New Brunswick, Canada;Department of Computer Science, University of British Columbia, Canada;School of Information Technologies, The University of Sydney, Australia;Department of Mathematics, Davidson College;Department of Computer Science, University of Arizona;Department of Computer Science, Williams College;Department of Computer Science, University of Perugia, Italy;School of Computing, Queens University, Canada;Department of Computer Science, University of Victoria, Canada
Venue:
GD'12 Proceedings of the 20th international conference on Graph Drawing
Year:
2012

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Abstract

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.