### 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 language | English (US) |
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Title of host publication | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Pages | 187-198 |

Number of pages | 12 |

Volume | 7704 LNCS |

DOIs | |

State | Published - 2013 |

Event | 20th International Symposium on Graph Drawing, GD 2012 - Redmond, WA, United States Duration: Sep 19 2012 → Sep 21 2012 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 7704 LNCS |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 20th International Symposium on Graph Drawing, GD 2012 |
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Country | United States |

City | Redmond, WA |

Period | 9/19/12 → 9/21/12 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)
- Theoretical Computer Science

### Cite this

*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.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*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

}

TY - GEN

T1 - On representing graphs by touching cuboids

AU - Bremner, David

AU - Evans, William

AU - Frati, Fabrizio

AU - Heyer, Laurie

AU - Kobourov, Stephen G

AU - Lenhart, William J.

AU - Liotta, Giuseppe

AU - Rappaport, David

AU - Whitesides, Sue H.

PY - 2013

Y1 - 2013

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84874144058&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84874144058&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-36763-2_17

DO - 10.1007/978-3-642-36763-2_17

M3 - Conference contribution

AN - SCOPUS:84874144058

SN - 9783642367625

VL - 7704 LNCS

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 187

EP - 198

BT - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

ER -