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(n2).


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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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