Advertisement

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(n 2)-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.

Keywords

Common Edge Outer Face Cycle Basis Expansion Graph Split Component 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Di Battista, G., Tamassia, R.: On-line maintenance of triconnected components with SPQR-trees. Algorithmica 15(4), 302–318 (1996)MathSciNetzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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

Personalised recommendations