An SPQR-Tree Approach to Decide Special Cases of Simultaneous Embedding with Fixed Edges

  • J. Joseph Fowler
  • Carsten Gutwenger
  • Michael Jünger
  • Petra Mutzel
  • Michael Schulz
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5417)


We present a linear-time algorithm for solving the simultaneous embedding problem with fixed edges (SEFE) for a planar graph and a pseudoforest (a graph with at most one cycle) by reducing it to the following embedding problem: Given a planar graph G, a cycle C of G, and a partitioning of the remaining vertices of G, does there exist a planar embedding in which the induced subgraph on each vertex partite of G ∖ C is contained entirely inside or outside C? For the latter problem, we present an algorithm that is based on SPQR-trees and has linear running time. We also show how we can employ SPQR-trees to decide SEFE for two planar graphs where one graph has at most two cycles and the intersection is a pseudoforest in linear time. These results give rise to our hope that our SPQR-tree approach might eventually lead to a polynomial-time algorithm for deciding the general SEFE problem for two planar graphs.


Planar Graph Decision Algorithm Cyclic Order Expansion Graph Auxiliary Graph 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • J. Joseph Fowler
    • 1
  • Carsten Gutwenger
    • 2
  • Michael Jünger
    • 3
  • Petra Mutzel
    • 2
  • Michael Schulz
    • 3
  1. 1.Department of Computer ScienceUniversity of ArizonaUSA
  2. 2.Department of Computer ScienceTechnische Universität DortmundGermany
  3. 3.Department of Computer ScienceUniversity of CologneGermany

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