Simultaneous Embeddability of Two Partitions

  • Jan Christoph Athenstädt
  • Tanja Hartmann
  • Martin Nöllenburg
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8871)


We study the simultaneous embeddability of a pair of partitions of the same underlying set into disjoint blocks. Each element of the set is mapped to a point in the plane and each block of either of the two partitions is mapped to a region that contains exactly those points that belong to the elements in the block and that is bounded by a simple closed curve. We establish three main classes of simultaneous embeddability (weak, strong, and full embeddability) that differ by increasingly strict well-formedness conditions on how different block regions are allowed to intersect. We show that these simultaneous embeddability classes are closely related to different planarity concepts of hypergraphs. For each embeddability class we give a full characterization. We show that (i) every pair of partitions has a weak simultaneous embedding, (ii) it is NP-complete to decide the existence of a strong simultaneous embedding, and (iii) the existence of a full simultaneous embedding can be tested in linear time.


Planar Graph Block Region Base Grid Full Embeddability Contour Graph 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Jan Christoph Athenstädt
    • 1
  • Tanja Hartmann
    • 2
  • Martin Nöllenburg
    • 2
  1. 1.Department of Computer and Information ScienceUniversity of KonstanzGermany
  2. 2.Institute of Theoretical InformaticsKarlsruhe Institute of Technology (KIT)Germany

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