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)


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.


Common Edge Outer Face Cycle Basis Expansion Graph Split Component 
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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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