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

10 Citations (Scopus)

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.

Original languageEnglish (US)
Title of host publicationLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Pages187-198
Number of pages12
Volume7704 LNCS
DOIs
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)03029743
ISSN (Electronic)16113349

Other

Other20th International Symposium on Graph Drawing, GD 2012
CountryUnited States
CityRedmond, WA
Period9/19/129/21/12

Fingerprint

Cuboid
Contact
Graph in graph theory
Unit cube
Regular hexahedron
Intersect
Planar graph
Rectangle
Disjoint
Interior
NP-complete problem
Intersection
Theorem

ASJC Scopus subject areas

  • Computer Science(all)
  • Theoretical Computer Science

Cite this

Bremner, D., Evans, W., Frati, F., Heyer, L., Kobourov, S. G., Lenhart, W. J., ... Whitesides, S. H. (2013). On representing graphs by touching cuboids. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 7704 LNCS, pp. 187-198). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 7704 LNCS). https://doi.org/10.1007/978-3-642-36763-2_17

On representing graphs by touching cuboids. / Bremner, David; Evans, William; Frati, Fabrizio; Heyer, Laurie; Kobourov, Stephen G; Lenhart, William J.; Liotta, Giuseppe; Rappaport, David; Whitesides, Sue H.

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Vol. 7704 LNCS 2013. p. 187-198 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 7704 LNCS).

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

Bremner, D, Evans, W, Frati, F, Heyer, L, Kobourov, SG, Lenhart, WJ, Liotta, G, Rappaport, D & Whitesides, SH 2013, On representing graphs by touching cuboids. in Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). vol. 7704 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 7704 LNCS, pp. 187-198, 20th International Symposium on Graph Drawing, GD 2012, Redmond, WA, United States, 9/19/12. https://doi.org/10.1007/978-3-642-36763-2_17
Bremner D, Evans W, Frati F, Heyer L, Kobourov SG, Lenhart WJ et al. On representing graphs by touching cuboids. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Vol. 7704 LNCS. 2013. p. 187-198. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-642-36763-2_17
Bremner, David ; Evans, William ; Frati, Fabrizio ; Heyer, Laurie ; Kobourov, Stephen G ; Lenhart, William J. ; Liotta, Giuseppe ; Rappaport, David ; Whitesides, Sue H. / On representing graphs by touching cuboids. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Vol. 7704 LNCS 2013. pp. 187-198 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
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