Disconnectivity and Relative Positions in Simultaneous Embeddings

  • Thomas Bläsius
  • Ignaz Rutter
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7704)


For two planar graph \(G^{\textcircled1}\) = (\(V^{\textcircled1}\), \(E^{\textcircled1}\)) and \(G^{\textcircled2}\) = (\(V^{\textcircled2}\), \(E^{\textcircled2}\)) sharing a common subgraph G = \(G^{\textcircled1}\)\(G^{\textcircled2}\) the problem Simultaneous Embedding with Fixed Edges (SEFE) asks whether they admit planar drawings such that the common graph is drawn the same. Previous algorithms only work for cases where G is connected, and hence do not need to handle relative positions of connected components. We consider the problem where G, \(G^{\textcircled1}\) and \(G^{\textcircled2}\) are not necessarily connected.

First, we show that a general instance of SEFE can be reduced in linear time to an equivalent instance where \(V^{\textcircled1}\) = \(V^{\textcircled2}\) and \(G^{\textcircled1}\) and \(G^{\textcircled2}\) are connected. Second, for the case where G consists of disjoint cycles, we introduce the CC-tree which represents all embeddings of G that extend to planar embeddings of \(G^{\textcircled1}\). We show that CC-trees can be computed in linear time, and that their intersection is again a CC-tree. This yields a linear-time algorithm for SEFE if all k input graphs (possibly k > 2) pairwise share the same set of disjoint cycles. These results, including the CC-tree, extend to the case where G consists of arbitrary connected components, each with a fixed embedding. Then the running time is O(n 2).


Planar Graph Input Graph Outer Face Disjoint Cycle Expansion Graph 


  1. 1.
    Angelini, P., Di Battista, G., Frati, F., Jelínek, V., Kratochvíl, J., Patrignani, M., Rutter, I.: Testing planarity of partially embedded graphs. In: Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, pp. 202–221. SIAM (2010)Google Scholar
  2. 2.
    Angelini, P., Di Battista, G., Frati, F., Patrignani, M., Rutter, I.: Testing the Simultaneous Embeddability of two Graphs whose Intersection is a Biconnected or a Connected Graph. J. Discr. Alg. 14, 150–172 (2012), http://dx.doi.org/10.1016/j.jda.2011.12.015 MATHCrossRefGoogle Scholar
  3. 3.
    Bläsius, T., Kobourov, S.G., Rutter, I.: Simultaneous embedding of planar graphs. CoRR abs/1204.5853 (2012)Google Scholar
  4. 4.
    Bläsius, T., Rutter, I.: Simultaneous PQ-Ordering with Applications to Constrained Embedding Problems. CoRR abs/1112.0245 (2011)Google Scholar
  5. 5.
    Di Battista, G., Tamassia, R.: On-Line Maintenance of Triconnected Components with SPQR-Trees. Algorithmica 15(4), 302–318 (1996)MathSciNetMATHGoogle Scholar
  6. 6.
    Di Battista, G., Tamassia, R.: On-Line Planarity Testing. SIAM J. Comput. 25(5), 956–997 (1996)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Fowler, J.J., Gutwenger, C., Jünger, M., Mutzel, P., Schulz, M.: An SPQR-Tree Approach to Decide Special Cases of Simultaneous Embedding with Fixed Edges. In: Tollis, I.G., Patrignani, M. (eds.) GD 2008. LNCS, vol. 5417, pp. 157–168. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  8. 8.
    Gassner, E., Jünger, M., Percan, M., Schaefer, M., Schulz, M.: Simultaneous Graph Embeddings with Fixed Edges. In: Fomin, F.V. (ed.) WG 2006. LNCS, vol. 4271, pp. 325–335. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  9. 9.
    Gutwenger, C., Mutzel, P.: A Linear Time Implementation of SPQR-Trees. In: Marks, J. (ed.) GD 2000. LNCS, vol. 1984, pp. 77–90. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  10. 10.
    Haeupler, B., Jampani, K.R., Lubiw, A.: Testing Simultaneous Planarity When the Common Graph Is 2-Connected. In: Cheong, O., Chwa, K.-Y., Park, K. (eds.) ISAAC 2010, Part II. LNCS, vol. 6507, pp. 410–421. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  11. 11.
    Jünger, M., Schulz, M.: Intersection Graphs in Simultaneous Embedding with Fixed Edges. Journal of Graph Algorithms and Applications 13(2), 205–218 (2009)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Whitney, H.: Congruent Graphs and the Connectivity of Graphs. American Journal of Mathematics 54(1), 150–168 (1932)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Thomas Bläsius
    • 1
  • Ignaz Rutter
    • 1
  1. 1.Karlsruhe Institute of Technology (KIT)Germany

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