Constrained Simultaneous and Near-Simultaneous Embeddings

  • Fabrizio Frati
  • Michael Kaufmann
  • Stephen G. Kobourov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4875)


A geometric simultaneous embedding of two graphs G 1 = (V 1,E 1) and G 2 = (V 2,E 2) with a bijective mapping of their vertex sets γ: V 1V 2 is a pair of planar straight-line drawings Γ 1 of G 1 and Γ 2 of G 2, such that each vertex v 2 = γ(v 1) is mapped in Γ 2 to the same point where v 1 is mapped in Γ 1, where v 1 ∈ V 1 and v 2 ∈ V 2.

In this paper we examine several constrained versions and a relaxed version of the geometric simultaneous embedding problem. We show that if the input graphs are assumed to share no common edges this does not seem to yield large classes of graphs that can be simultaneously embedded. Further, if a prescribed combinatorial embedding for each input graph must be preserved, then we can answer some of the problems that are still open for geometric simultaneous embedding. Finally, we present some positive and negative results on the near-simultaneous embedding problem, in which vertices are not mapped exactly to the same but to “near” points in the different drawings.


  1. 1.
    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. Comput. Geom. 36(2), 117–130 (2007)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Collberg, C., Kobourov, S.G., Nagra, J., Pitts, J., Wampler, K.: A system for graph-based visualization of the evolution of software. In: SoftVis, pp. 377–386 (2003)Google Scholar
  3. 3.
    Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing. Prentice Hall, Upper Saddle River, NJ (1999)MATHCrossRefGoogle Scholar
  4. 4.
    Di Giacomo, E., Liotta, G.: A note on simultaneous embedding of planar graphs. In: EWCG, pp. 207–210 (2005)Google Scholar
  5. 5.
    Erten, C., Kobourov, S.G.: Simultaneous embedding of planar graphs with few bends. In: Pach, J. (ed.) GD 2004. LNCS, vol. 3383, pp. 195–205. Springer, Heidelberg (2005)Google Scholar
  6. 6.
    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
  7. 7.
    Frati, F., Kaufmann, M., Kobourov, S.: Constrained simultaneous and near-simultaneous embeddings. Tech. Report RT-DIA-120-2007, University of Roma Tre (2007),
  8. 8.
    Geyer, M., Kaufmann, M., Vrto, 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
  9. 9.
    Healy, P., Kuusik, A., Leipert, S.: A characterization of level planar graphs. Discr. Math. 280(1-3), 51–63 (2004)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Jünger, M., Leipert, S.: Level planar embedding in linear time. Jour. of Graph Alg. and Appl. 6(1), 67–113 (2002)MATHGoogle Scholar
  11. 11.
    Kaufmann, M., Wagner, D. (eds.): Drawing Graphs. LNCS, vol. 2025. Springer, Heidelberg (2001)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Fabrizio Frati
    • 1
  • Michael Kaufmann
    • 2
  • Stephen G. Kobourov
    • 3
  1. 1.Dipartimento di Informatica e AutomazioneUniversità Roma TreItaly
  2. 2.Wilhelm-Schickard-Institut für InformatikUniversität TübingenGermany
  3. 3.Department of Computer ScienceUniversity of ArizonaU.S.A.

Personalised recommendations