Simultaneous Embedding: Edge Orderings, Relative Positions, Cutvertices

  • Thomas Bläsius
  • Annette Karrer
  • Ignaz Rutter
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8242)

Abstract

A simultaneous embedding of two graphs \(G^{\mbox{\textcircled{1}}}\) and \(G^{\mbox{\textcircled{2}}}\) with common graph \(G=G^{\mbox{\textcircled{1}}} \cap G^{\mbox{\textcircled{2}}}\) is a pair of planar drawings of \(G^{\mbox{\textcircled{1}}}\) and \(G^{\mbox{\textcircled{2}}}\) that coincide on G. It is an open question whether there is a polynomial-time algorithm that decides whether two graphs admit a simultaneous embedding (problem Sefe).

In this paper, we present two results. First, a set of three linear-time preprocessing algorithms that remove certain substructures from a given Sefe instance, producing a set of equivalent Sefe instances without such substructures. The structures we can remove are (1) cutvertices of the union graph\(G^{\mbox{\textcircled{1}}} \cup G^{\mbox{\textcircled{2}}}\), (2) cutvertices that are simultaneously a cutvertex in \(G^{\mbox{\textcircled{1}}}\) and \(G^{\mbox{\textcircled{2}}}\) and that have degree at most 3 in G, and (3) connected components of G that are biconnected but not a cycle.

Second, we give an O(n2)-time algorithm for Sefe where, for each pole u of a P-node μ (of a block) of the input graphs, at most three virtual edges of μ contain common edges incident to u. All algorithms extend to the sunflower case.

References

  1. 1.
    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. Journal of Discrete Algorithms 14, 150–172 (2012)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bläsius, T., Kobourov, S.G., Rutter, I.: Simultaneous Embedding of Planar Graphs. In: Handbook of Graph Drawing and Visualization, pp. 349–381. Chapman and Hall/CRC (2013)Google Scholar
  3. 3.
    Bläsius, T., Rutter, I.: Disconnectivity and relative positions in simultaneous embeddings. In: Didimo, W., Patrignani, M. (eds.) GD 2012. LNCS, vol. 7704, pp. 31–42. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  4. 4.
    Bläsius, T., Rutter, I.: Simultaneous PQ-ordering with applications to constrained embedding problems. In: Proc. 24th ACM-SIAM Sympos. Discrete Algorithm, SODA 2013, pp. 1030–1043. ACM (2013)Google Scholar
  5. 5.
    Booth, K.S., Lueker, G.S.: Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms. Journal of Computer and System Sciences 13, 335–379 (1976)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Di Battista, G., Tamassia, R.: On-line maintenance of triconnected components with SPQR-trees. Algorithmica 15(4), 302–318 (1996)MathSciNetMATHGoogle Scholar
  7. 7.
    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
  8. 8.
    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
  9. 9.
    Haeupler, B., Jampani, K., Lubiw, A.: Testing simultaneous planarity when the common graph is 2-connected. J. Graph Algorithms Appl. 17(3), 147–171 (2013)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Jünger, M., Schulz, M.: Intersection graphs in simultaneous embedding with fixed edges. Journal of Graph Algorithms and Applications 13(2), 205–218 (2009)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Schaefer, M.: Toward a theory of planarity: Hanani-tutte and planarity variants. In: Didimo, W., Patrignani, M. (eds.) GD 2012. LNCS, vol. 7704, pp. 162–173. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  12. 12.
    Wiedemann, D.: Solving sparse linear equations over finite fields. IEEE Transactions on Information Theory 32(1), 54–62 (1986)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

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

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