Edge-Weighted Contact Representations of Planar Graphs

  • Martin Nöllenburg
  • Roman Prutkin
  • Ignaz Rutter
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7704)


We study contact representations of edge-weighted planar graphs, where vertices are rectangles or rectilinear polygons and edges are polygon contacts whose lengths represent the edge weights. We show that for any given edge-weighted planar graph whose outer face is a quadrangle, that is internally triangulated and that has no separating triangles we can construct in linear time an edge-proportional rectangular dual if one exists and report failure otherwise. For a given combinatorial structure of the contact representation and edge weights interpreted as lower bounds on the contact lengths, a corresponding contact representation that minimizes the size of the enclosing rectangle can be found in linear time.If the combinatorial structure is not fixed, we prove NP-hardness of deciding whether a contact representation with bounded contact lengths exists.

Finally, we give a complete characterization of the rectilinear polygon complexity required for representing biconnected internally triangulated graphs: For outerplanar graphs complexity 8 is sufficient and necessary, and for graphs with two adjacent or multiple non-adjacent internal vertices the complexity is unbounded.


Planar Graph Edge Weight Contact Length Internal Vertex Outer Face 
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.


  1. 1.
    Alam, M.J., Biedl, T., Felsner, S., Gerasch, A., Kaufmann, M., Kobourov, S.G.: Linear-Time Algorithms for Hole-Free Rectilinear Proportional Contact Graph Representations. In: Asano, T., Nakano, S.-i., Okamoto, Y., Watanabe, O. (eds.) ISAAC 2011. LNCS, vol. 7074, pp. 281–291. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  2. 2.
    Alam, M.J., Biedl, T., Felsner, S., Kaufmann, M., Kobourov, S.G., Ueckerdt, T.: Computing cartograms with optimal complexity. In: Proc. 28th ACM Symp. on Computational Geometry, SoCG 2012, pp. 21–30. ACM (2012)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.
    de Berg, M., Mumford, E., Speckmann, B.: On rectilinear duals for vertex-weighted plane graphs. Discrete Mathematics 309(7), 1794–1812 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Biedl, T., Genc, B.: Complexity of octagonal and rectangular cartograms. In: Proc. 17th Canadian Conference on Computational Geometry, CCCG 2005, pp. 117–120 (2005)Google Scholar
  6. 6.
    Biedl, T., Ruiz Velázquez, L.E.: Orthogonal Cartograms with Few Corners Per Face. In: Dehne, F., Iacono, J., Sack, J.-R. (eds.) WADS 2011. LNCS, vol. 6844, pp. 98–109. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  7. 7.
    Cabello, S., Demaine, E.D., Rote, G.: Planar embeddings of graphs with specified edge lengths. J. Graph Algorithms Appl. 11(1), 259–276 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Chalopin, J., Gonçalves, D.: Every planar graph is the intersection graph of segments in the plane. In: Proc. 41st Ann. ACM Symp. Theory of Computing, STOC 2009, pp. 631–638. ACM (2009)Google Scholar
  9. 9.
    Eades, P., Wormald, N.C.: Fixed edge-length graph drawing is NP-hard. Discrete Applied Mathematics 28(2), 111–134 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Eppstein, D., Mumford, E., Speckmann, B., Verbeek, K.: Area-universal and constrained rectangular layouts. SIAM J. Comput. 41(3), 537–564 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    He, X.: On finding the rectangular duals of planar triangular graphs. SIAM J. Comput. 22(6), 1218–1226 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Hliněný, P., Kratochvíl, J.: Representing graphs by disks and balls (a survey of recognition-complexity results). Discrete Mathematics 229(1-3), 101–124 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Kawaguchi, A., Nagamochi, H.: Orthogonal Drawings for Plane Graphs with Specified Face Areas. In: Cai, J.-Y., Cooper, S.B., Zhu, H. (eds.) TAMC 2007. LNCS, vol. 4484, pp. 584–594. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  14. 14.
    Knuth, D.E., Raghunathan, A.: The problem of compatible representatives. SIAM J. Discrete Math. 5(3), 422–427 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Koebe, P.: Kontaktprobleme der konformen abbildung. Ber. Sächs. Akad. Wiss. Leipzig, Math.-Phys. Klasse 88, 141–164 (1936)Google Scholar
  16. 16.
    Kozminski, K.A., Kinnen, E.: Rectangular dualization and rectangular dissections. IEEE Trans. Circuits and Systems 35(11), 1401–1416 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    van Kreveld, M., Speckmann, B.: On rectangular cartograms. Comput. Geom. Theory Appl. 37(3), 175–187 (2007)zbMATHCrossRefGoogle Scholar
  18. 18.
    Leinwand, S.M., Lai, Y.T.: An algorithm for building rectangular floor-plans. In: Proc. 21st Design Automation Conference, pp. 663–664 (1984)Google Scholar
  19. 19.
    Liao, C.C., Lu, H.I., Yen, H.C.: Compact floor-planning via orderly spanning trees. J. Algorithms 48(2), 441–451 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Lichtenstein, D.: Planar formulae and their uses. SIAM J. Comput. 11(2), 329–343 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    McKee, T.A., McMorris, F.R.: Topics in Intersection Graph Theory. SIAM (1999)Google Scholar
  22. 22.
    Nishizeki, T., Rahman, M.S.: Rectangular drawing algorithms. In: Tamassia, R. (ed.) Handbook of Graph Drawing and Visualization, ch. 13. CRC Press (2013) (to appear)Google Scholar
  23. 23.
    Tamassia, R.: Drawing algorithms for planar st-graphs. Australasian J. Combinatorics 2, 217–235 (1990)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Yeap, K.H., Sarrafzadeh, M.: Floor-planning by graph dualization: 2-concave rectilinear modules. SIAM J. Comput. 22(3), 500–526 (1993)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Martin Nöllenburg
    • 1
  • Roman Prutkin
    • 1
  • Ignaz Rutter
    • 1
  1. 1.Institut für Theoretische InformatikKarlsruhe Institute of TechnologyGermany

Personalised recommendations