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

Research output: Chapter in Book/Report/Conference proceedingConference contribution

11 Scopus citations


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.

Original languageEnglish (US)
Title of host publicationGraph Drawing - 20th International Symposium, GD 2012, Revised Selected Papers
Number of pages12
StatePublished - 2013
Event20th International Symposium on Graph Drawing, GD 2012 - Redmond, WA, United States
Duration: Sep 19 2012Sep 21 2012

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume7704 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Other20th International Symposium on Graph Drawing, GD 2012
Country/TerritoryUnited States
CityRedmond, WA

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)


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