Matched Drawings of Planar Graphs

  • Emilio Di Giacomo
  • Walter Didimo
  • Marc van Kreveld
  • Giuseppe Liotta
  • Bettina Speckmann
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4875)

Abstract

A natural way to draw two planar graphs whose vertex sets are matched is to assign each matched pair a unique y-coordinate. In this paper we introduce the concept of such matched drawings, which are a relaxation of simultaneous geometric embeddings with mapping. We study which classes of graphs allow matched drawings and show that (i) two 3-connected planar graphs or a 3-connected planar graph and a tree may not be matched drawable, while (ii) two trees or a planar graph and a planar graph of some special families—such as unlabeled level planar (ULP) graphs or the family of “carousel graphs”—are always matched drawable.

References

  1. 1.
    Brandes, U., Erlebach, T. (eds.): Network Analysis. LNCS, vol. 3418. Springer, Heidelberg (2005)MATHGoogle Scholar
  2. 2.
    Brandes, U., Erten, C., Fowler, J., Frati, F., Geyer, M., Gutwenger, C., Hong, S.-H., Kaufmann, M., Kobourov, S., Liotta, G., Mutzel, P., Symvonis, A.: Colored simultaneous geometric embeddings. In: Lin, G. (ed.) COCOON 2007. LNCS, vol. 4598, pp. 254–263. Springer, Heidelberg (2007)Google Scholar
  3. 3.
    Braß, P., Cenek, E., Duncan, C.A., Efrat, A., Erten, C., Ismailescu, D., Kobourov, S.G., Lubiw, A., Mitchell, J.S.B.: On simultaneous planar graph embeddings. Computational Geometry: Theory and Applications 36(2), 117–130 (2007)MATHMathSciNetGoogle Scholar
  4. 4.
    Cappos, J., Estrella-Balderrama, A., Fowler, J.J., Kobourov, S.G.: Simultaneous graph embedding with bends and circular arcs. In: Kaufmann, M., Wagner, D. (eds.) GD 2006. LNCS, vol. 4372, pp. 95–107. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  5. 5.
    de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10, 41–51 (1990)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Demetrescu, C., Di Battista, G., Finocchi, I., Liotta, G., Patrignani, M., Pizzonia, M.: Infinite trees and the future. In: Kratochvíl, J. (ed.) GD 1999. LNCS, vol. 1731, pp. 379–391. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  7. 7.
    Di Giacomo, E., Didimo, W., Grilli, L., Liotta, G.: Graph visualization techniques for web clustering engines. IEEE Transactions on Visualization and Computer Graphics 13(2), 294–304 (2007)CrossRefGoogle Scholar
  8. 8.
    Di Giacomo, E., Liotta, G.: Simultaneous embedding of outerplanar graphs, paths, and cycles. International Journal of Computational Geometry and Applications 17(2), 139–160 (2007)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Erten, C., Kobourov, S.G.: Simultaneous embedding of planar graphs with few bends. Journal of Graph Algorithms and Applications 9(3), 347–364 (2005)MathSciNetGoogle Scholar
  10. 10.
    Estrella-Balderrama, A., Fowler, J.J., Kobourov, S.G.: Characterization of unlabeled level planar trees. In: Kaufmann, M., Wagner, D. (eds.) GD 2006. LNCS, vol. 4372, pp. 367–379. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  11. 11.
    Fernau, H., Kaufmann, M., Poths, M.: Comparing trees via crossing minimization. In: Proc. 25th Conf. on Foundations of Software Technology and Theoretical Computer Science, pp. 457–469 (2005)Google Scholar
  12. 12.
    Fowler, J.J., Kouborov, S.G.: Characterization of unlabeled level planar graphs. Technical Report TR06-04, Dep. of Computer Science, University of Arizona (2006)Google Scholar
  13. 13.
    Frati, F.: Embedding graphs simultaneously with fixed edges. In: Kaufmann, M., Wagner, D. (eds.) GD 2006. LNCS, vol. 4372, pp. 108–113. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  14. 14.
    Friedrich, C., Eades, P.: Graph drawing in motion. Journal of Graph Algorithms and Applications 6(3), 353–370 (2002)MATHMathSciNetGoogle Scholar
  15. 15.
    Geyer, M., Kaufmann, M., Vrt’o, I.: Two trees which are self-intersecting when drawn simultaneously. In: Healy, P., Nikolov, N.S. (eds.) GD 2005. LNCS, vol. 3843, pp. 201–210. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  16. 16.
    Huang, M.L., Eades, P., Cohen, R.F.: WebOFDAV-navigating and visualising the web online with animated context swapping. Computer Networks and ISDN Systems 30, 638–642 (1998)CrossRefGoogle Scholar
  17. 17.
    Misue, K., Eades, P., Lai, W., Sugiyama, K.: Layout adjustment and the mental map. Journal of Visual Languages and Computing 6(2), 183–210 (1995)CrossRefGoogle Scholar
  18. 18.
    North, S.: Incremental layout in dynadag. In: Brandenburg, F.J. (ed.) GD 1995. LNCS, vol. 1027, pp. 409–418. Springer, Heidelberg (1996)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Emilio Di Giacomo
    • 1
  • Walter Didimo
    • 1
  • Marc van Kreveld
    • 2
  • Giuseppe Liotta
    • 1
  • Bettina Speckmann
    • 3
  1. 1.Dip. di Ingegneria Elettronica e dell’InformazioneUniversità degli Studi di Perugia 
  2. 2.Department of Computer ScienceUtrecht University 
  3. 3.Department of Mathematics and Computer ScienceTU Eindhoven 

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