Testing the Simultaneous Embeddability of Two Graphs Whose Intersection Is a Biconnected Graph or a Tree

  • Patrizio Angelini
  • Giuseppe Di Battista
  • Fabrizio Frati
  • Maurizio Patrignani
  • Ignaz Rutter
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6460)


In this paper we study the time complexity of the problem Simultaneous Embedding with Fixed Edges (Sefe), that takes two planar graphs G 1 = (V,E 1) and G 2 = (V,E 2) as input and asks whether a planar drawing Γ1 of G 1 and a planar drawing Γ2 of G 2 exist such that: (i) each vertex v ∈ V is mapped to the same point in Γ1 and in Γ2; (ii) every edge e ∈ E 1 ∩ E 2 is mapped to the same Jordan curve in Γ1 and Γ2. First, we show a polynomial-time algorithm for Sefe when the intersection graph of G 1 and G 2, that is the planar graph G 1 ∩ 2 = (V,E 1 ∩ E 2), is biconnected. Second, we show that Sefe, when G 1 ∩ 2 is a tree, is equivalent to a suitably-defined book embedding problem. Based on such an equivalence and on recent results by Hong and Nagamochi, we show a linear-time algorithm for the Sefe problem when G 1 ∩ 2 is a star.


Outer Edge Jordan Curve Intersection Graph Internal Vertex Positive Instance 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Patrizio Angelini
    • 1
  • Giuseppe Di Battista
    • 1
  • Fabrizio Frati
    • 1
  • Maurizio Patrignani
    • 1
  • Ignaz Rutter
    • 2
  1. 1.Dipartimento di Informatica e AutomazioneUniversità Roma TreItaly
  2. 2.Institute of Theoretical InformaticsKarlsruhe Institute of Technology (KIT)Germany

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