GD 2012: Graph Drawing pp 187-198

# 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
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7704)

## 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.

## Keywords

Planar Graph Unit Cube Edge Contact Face Contact Contact Graph

## References

1. 1.
Alam, M.J., Biedl, T.C., Felsner, S., Kaufmann, M., Kobourov, S.G., Ueckerdt, T.: Computing cartograms with optimal complexity. In: Dey, T.K., Whitesides, S. (eds.) Symposium on Computational Geometry, SoCG 2012, pp. 21–30 (2012)Google Scholar
2. 2.
Badent, M., Binucci, C., Giacomo, E.D., Didimo, W., Felsner, S., Giordano, F., Kratochvíl, J., Palladino, P., Patrignani, M., Trotta, F.: Homothetic triangle contact representations of planar graphs. In: Bose, P. (ed.) Canadian Conference on Computational Geometry, CCCG 2007, pp. 233–236 (2007)Google Scholar
3. 3.
Battista, G.D., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing: Algorithms for the Visualization of Graphs. Prentice-Hall (1999)Google Scholar
4. 4.
Borchard-Ott, W.: Crystallography. Springer (2011)Google Scholar
5. 5.
Breu, H.: Algorithmic Aspects of Constrained Unit Disk Graphs. Ph.D. thesis, The University of British Columbia, Canada (1996)Google Scholar
6. 6.
Buchsbaum, A.L., Gansner, E.R., Procopiuc, C.M., Venkatasubramanian, S.: Rectangular layouts and contact graphs. ACM Transactions on Algorithms 4(1) (2008)Google Scholar
7. 7.
Czyzowicz, J., Kranakis, E., Krizanc, D., Urrutia, J.: Discrete realizations of contact and intersection graphs. International Journal of Pure and Applied Mathematics 13(4), 429–442 (2004)
8. 8.
de Fraysseix, H., Ossona de Mendez, P.: Representations of Planar Graphs by Segments. Colloquia Mathematica Societatis János Bolyai, vol. 63, pp. 109–117. North-Holland (2007)Google Scholar
9. 9.
Duncan, C.A., Gansner, E.R., Hu, Y.F., Kaufmann, M., Kobourov, S.G.: Optimal polygonal representation of planar graphs. Algorithmica 63(3), 672–691 (2012)
10. 10.
Eades, P., Whitesides, S.: The logic engine and the realization problem for nearest neighbor graphs. Theoretical Computer Science 169(1), 23–37 (1996)
11. 11.
Felsner, S.: Rectangle and square representations of planar graphs. In: Pach, J. (ed.) Thirty Essays on Geometric Graph Theory. Springer (2012)Google Scholar
12. 12.
Felsner, S., Francis, M.C.: Contact representations of planar graphs with cubes. In: Hurtado, F., van Kreveld, M.J. (eds.) Symposium on Computational Geometry, SoCG 2011, pp. 315–320 (2011)Google Scholar
13. 13.
de Fraysseix, H., Ossona de Mendez, P., Rosenstiehl, P.: On triangle contact graphs. Combinatorics, Probability & Computing 3, 233–246 (1994)
14. 14.
de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10(1), 41–51 (1990)
15. 15.
He, X.: On floor-plan of plane graphs. SIAM Journal on Computing 28(6), 2150–2167 (1999)
16. 16.
Kaufmann, M., Wagner, D. (eds.): Drawing graphs: methods and models. Springer (2001)Google Scholar
17. 17.
Koebe, P.: Kontaktprobleme der konformen Abbildung. Berichte über die Verhandlungen der Sächsischen Akad. der Wissenschaften zu Leipzig. Math.-Phys. Klasse 88, 141–164 (1936)Google Scholar
18. 18.
Koźmiński, K., Kinnen, E.: Rectangular duals of planar graphs. Networks 15, 145–157 (1985)
19. 19.
Leinwand, S.M., Lai, Y.T.: An algorithm for building rectangular floor-plans. In: Lambert, P.H., Ofek, H., O’Neill, L.A., Pistilli, P.O., Losleben, P., Nash, J.D., Shaklee, D.W., Preas, B.T., Lerman, H.N. (eds.) Design Automation Conference, DAC 1984, pp. 663–664 (1984)Google Scholar
20. 20.
Schnyder, W.: Planar graphs and poset dimension. Order 5(4), 323–343 (1989)
21. 21.
Schnyder, W.: Embedding planar graphs on the grid. In: Symposium on Discrete Algorithms, SODA 1990, pp. 138–148 (1990)Google Scholar
22. 22.
Thomassen, C.: Interval representations of planar graphs. Journal of Combinatorial Theory, Series B 40(1), 9–20 (1986)
23. 23.
Ungar, P.: On diagrams representing graphs. Journal of the London Mathematical Society 28, 336–342 (1953)

## Authors and Affiliations

• David Bremner
• 1
• William Evans
• 2
• Fabrizio Frati
• 3
• Laurie Heyer
• 4
• Stephen G. Kobourov
• 5
• William J. Lenhart
• 6
• Giuseppe Liotta
• 7
• David Rappaport
• 8
• Sue H. Whitesides
• 9
1. 1.Faculty of Computer ScienceUniversity of New BrunswickCanada
2. 2.Department of Computer ScienceUniversity of British ColumbiaCanada
3. 3.School of Information TechnologiesThe University of SydneyAustralia
4. 4.Department of MathematicsDavidson CollegeUSA
5. 5.Department of Computer ScienceUniversity of ArizonaUSA
6. 6.Department of Computer ScienceWilliams CollegeUSA
7. 7.Department of Computer ScienceUniversity of PerugiaItaly