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Towards Characterizing Graphs with a Sliceable Rectangular Dual

  • Vincent KustersEmail author
  • Bettina Speckmann
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9411)

Abstract

Let \(\mathcal {G} \) be a plane triangulated graph. A rectangular dual of \(\mathcal {G} \) is a partition of a rectangle R into a set \(\mathcal {R} \) of interior-disjoint rectangles, one for each vertex, such that two regions are adjacent if and only if the corresponding vertices are connected by an edge. A rectangular dual is sliceable if it can be recursively subdivided along horizontal or vertical lines. A graph is rectangular if it has a rectangular dual and sliceable if it has a sliceable rectangular dual. There is a clear characterization of rectangular graphs. However, a full characterization of sliceable graphs is still lacking. The currently best result (Yeap and Sarrafzadeh, 1995) proves that all rectangular graphs without a separating 4-cycle are sliceable. In this paper we introduce a recursively defined class of graphs and prove that these graphs are precisely the nonsliceable graphs with exactly one separating 4-cycle.

Keywords

Interior Vertex Blue Edge Vertical Slice Construction Sequence Extended Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Ackerman, E., Barequet, G., Pinter, R.Y., Romik, D.: The number of guillotine partitions in d dimensions. Inf. Proces. Letters 98(4), 162–167 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alam, M.J., Biedl, T., Felsner, S., Kaufmann, M., Kobourov, S.G., Ueckerdt, T.: Computing cartograms with optimal complexity. In: SOCG 2012, pp. 21–30 (2012)Google Scholar
  3. 3.
    de Berg, M., Mumford, E., Speckmann, B.: On rectilinear duals for vertex-weighted plane graphs. Disc. Math. 309(7), 1794–1812 (2009)CrossRefzbMATHGoogle Scholar
  4. 4.
    Bhasker, J., Sahni, S.: A linear time algorithm to check for the existence of a rectangular dual of a planar triangulated graph. Networks 17(3), 307–317 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bhattacharya, B., Sur-Kolay, S.: On the family of inherently nonslicible floorplans in VLSI layout design. In: ISCAS 1991, pp. 2850–2853. IEEE (1991)Google Scholar
  6. 6.
    Buchin, K., Speckmann, B., Verdonschot, S.: Optimizing regular edge labelings. In: Brandes, U., Cornelsen, S. (eds.) GD 2010. LNCS, vol. 6502, pp. 117–128. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  7. 7.
    Dasgupta, P., Sur-Kolay, S.: Slicible rectangular graphs and their optimal floorplans. ACM Trans. Design Automation of Electronic Systems 6(4), 447–470 (2001)Google Scholar
  8. 8.
    Eppstein, D., Mumford, E., Speckmann, B., Verbeek, K.: Area-universal rectangular layouts. In: SOCG 2009, pp. 267–276 (2009)Google Scholar
  9. 9.
    Fusy, É.: Transversal structures on triangulations: A combinatorial study and straight-line drawings. Disc. Math. 309(7), 1870–1894 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    He, X.: On floor-plan of plane graphs. SIAM J. Comp. 28(6), 2150–2167 (1999)CrossRefzbMATHGoogle Scholar
  11. 11.
    Kant, G., He, X.: Two algorithms for finding rectangular duals of planar graphs. In: van Leeuwen, J. (ed.) WG 1993. LNCS, vol. 790, pp. 396–410. Springer, Heidelberg (1994)Google Scholar
  12. 12.
    Koźmiński, K., Kinnen, E.: Rectangular duals of planar graphs. Networks 15(2), 145–157 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    van Kreveld, M., Speckmann, B.: On rectangular cartograms. Comp. Geom. 37(3), 175–187 (2007)CrossRefzbMATHGoogle Scholar
  14. 14.
    Liao, C.C., Lu, H.I., Yen, H.C.: Compact floor-planning via orderly spanning trees. J. Algorithms 48(2), 441–451 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Mumford, E.: Drawing Graphs for Cartographic Applications. Ph.D. thesis, TU Eindhoven (2008). http://repository.tue.nl/636963
  16. 16.
    Otten, R.: Efficient floorplan optimization. In: ICCAD’83. vol. 83, pp. 499–502 (1983)Google Scholar
  17. 17.
    Stockmeyer, L.: Optimal orientations of cells in slicing floorplan designs. Inf. Control 57(2), 91–101 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Sur-Kolay, S., Bhattacharya, B.: Inherent nonslicibility of rectangular duals in VLSI floorplanning. In: Kumar, S., Nori, K.V. (eds.) FSTTCS 1988. LNCS, vol. 338, pp. 88–107. Springer, Heidelberg (1988) CrossRefGoogle Scholar
  19. 19.
    Szepieniec, A.A., Otten, R.H.: The genealogical approach to the layout problem. In: Proceedings of the 17th Conference on Design Automation, pp. 535–542. IEEE (1980)Google Scholar
  20. 20.
    Ungar, P.: On diagrams representing maps. J. L. Math. Soc. 1(3), 336–342 (1953)CrossRefGoogle Scholar
  21. 21.
    Yao, B., Chen, H., Cheng, C.K., Graham, R.: Floorplan representations: Complexity and connections. ACM Trans. Design Auto. of Elec. Sys. 8(1), 55–80 (2003)Google Scholar
  22. 22.
    Yeap, G., Sarrafzadeh, M.: Sliceable floorplanning by graph dualization. SIAM J. Disc. Math. 8(2), 258–280 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Yeap, K.H., Sarrafzadeh, M.: Floor-planning by graph dualization: 2-concave rectilinear modules. SIAM J. Comp. 22(3), 500–526 (1993)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Computer ScienceETH ZürichZürichSwitzerland
  2. 2.Department of Computer ScienceTU EindhovenEindhovenThe Netherlands

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