Fitting Planar Graphs on Planar Maps

  • Md. Jawaherul Alam
  • Michael Kaufmann
  • Stephen G. Kobourov
  • Tamara Mchedlidze
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8327)

Abstract

Graph and cartographic visualization have the common objective to provide intuitive understanding of some underlying data. We consider a problem that combines aspects of both by studying the problem of fitting planar graphs on planar maps. After providing an NP-hardness result for the general decision problem, we identify sufficient conditions so that a fit is possible on a map with rectangular regions. We generalize our techniques to non-convex rectilinear polygons, where we also address the problem of efficient distribution of the vertices inside the map regions.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Angelini, P., Frati, F., Kaufmann, M.: Straight-line rectangular drawings of clustered graphs. In: Symposium on Algorithms & Data Structures (WADS), pp. 25–36 (2009)Google Scholar
  2. 2.
    Battista, G.D., Drovandi, G., Frati, F.: How to draw a clustered tree. Journal of Discrete Algorithms 7(4), 479–499 (2009)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Battista, G.D., Frati, F.: Efficient c-planarity testing for embedded flat clustered graphs with small faces. Journal of Graph Algorithms and Applications 13(3), 349–378 (2009)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Bern, M.W., Gilbert, J.R.: Drawing the planar dual. Information Processing Letters 43(1), 7–13 (1992)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Bruls, M., Huizing, K., van Wijk, J.J.: Squarified treemaps. In: Joint Eurographics and IEEE TCVG Symposium on Visualization, pp. 33–42 (2000)Google Scholar
  6. 6.
    Buchsbaum, A., Gansner, E., Procopiuc, C., Venkatasubramanian, S.: Rectangular layouts and contact graphs. ACM Transactions on Algorithms 4(1) (2008)Google Scholar
  7. 7.
    Chambers, E., Eppstein, D., Goodrich, M., Löffler, M.: Drawing graphs in the plane with a prescribed outer face and polynomial area. Journal of Graph Algorithms and Applications 16(2), 243–259 (2012)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Coffey, J.W., Hoffman, R.R., Cañas, A.J.: Concept map-based knowledge modeling: perspectives from information and knowledge visualization. Information Visualization 5(3), 192–201 (2006)CrossRefGoogle Scholar
  9. 9.
    Duncan, C., Gansner, E., Hu, Y., Kaufmann, M., Kobourov, S.G.: Optimal polygonal representation of planar graphs. Algorithmica 63(3), 672–691 (2012)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Eades, P., Feng, Q.-W., Lin, X., Nagamochi, H.: Straight-line drawing algorithms for hierarchical graphs and clustered graphs. Algorithmica 44(1), 1–32 (2006)CrossRefMATHMathSciNetGoogle Scholar
  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.
    Feng, Q.-W., Cohen, R.F., Eades, P.: How to draw a planar clustered graph. In: Li, M., Du, D.-Z. (eds.) COCOON 1995. LNCS, vol. 959, pp. 21–30. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  13. 13.
    Feng, Q.-W., Cohen, R.F., Eades, P.: Planarity for clustered graphs. In: European Symposium on Algorithms (ESA), pp. 213–226 (1995)Google Scholar
  14. 14.
    Fusy, É.: Transversal structures on triangulations: A combinatorial study and straight-line drawings. Discrete Mathematics 309(7), 1870–1894 (2009)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Harel, D.: On visual formalisms. Communications of the ACM 31(5), 514–530 (1988)CrossRefMathSciNetGoogle Scholar
  16. 16.
    He, X.: On floor-plan of plane graphs. SIAM Journal of Computing 28(6), 2150–2167 (1999)CrossRefMATHGoogle Scholar
  17. 17.
    Hu, Y., Gansner, E.R., Kobourov, S.G.: Visualizing graphs and clusters as maps. IEEE Computer Graphics and Applications 30(6), 54–66 (2010)CrossRefGoogle Scholar
  18. 18.
    Kamada, T.: Visualizing Abstract Objects and Relations. World Scientific Series in Computer Science, vol. 5 (1989)Google Scholar
  19. 19.
    Kerber, M.: Embedding the dual complex of hyper-rectangular partitions. Journal of Computational Geometry 4(1), 13–37 (2013)MathSciNetGoogle Scholar
  20. 20.
    Knuth, D.E., Raghunathan, A.: The problem of compatible representatives. SIAM Journal on Discrete Mathematics 5(3), 422–427 (1992)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Koźmiński, K., Kinnen, E.: Rectangular duals of planar graphs. Networks 15, 145–157 (1985)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Liao, C.-C., Lu, H.-I., Yen, H.-C.: Compact floor-planning via orderly spanning trees. Journal of Algorithms 48, 441–451 (2003)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Lichtenstein, D.: Planar formulae and their uses. SIAM Journal on Computing 11(2), 329–343 (1982)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Shneiderman, B.: Tree visualization with tree-maps: 2-D space-filling approach. ACM Transactions on Graphics 11(1), 92–99 (1992)CrossRefMATHGoogle Scholar
  25. 25.
    Ungar, P.: On diagrams representing graphs. Journal of London Mathematical Sociery 28, 336–342 (1953)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Yeap, K.-H., Sarrafzadeh, M.: Floor-planning by graph dualization: 2-concave rectilinear modules. SIAM Journal on Computing 22, 500–526 (1993)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Md. Jawaherul Alam
    • 1
  • Michael Kaufmann
    • 2
  • Stephen G. Kobourov
    • 1
  • Tamara Mchedlidze
    • 3
  1. 1.Department of Computer ScienceUniversity of ArizonaUSA
  2. 2.Institute for InformaticsUniversity of TübingenGermany
  3. 3.Institute for Theoretical InformaticsKarlsruhe Institute of TechnologyGermany

Personalised recommendations