Simultaneous Embeddability of Two Partitions
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.
KeywordsPlanar Graph Block Region Base Grid Full Embeddability Contour Graph
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- 1.Athenstädt, J.C., Hartmann, T., Nöllenburg, M.: Simultaneous embeddability of two partitions. CoRR, abs/1408.6019 (August 2014)Google Scholar
- 2.Bläsius, T., Kobourov, S.G., Rutter, I.: Simultaneous embedding of planar graphs. In: Tamassia, R. (ed.) Handbook of Graph Drawing and Visualization, ch. 11, pp. 349–381. CRC Press (2013)Google Scholar
- 5.Buchin, K., van Kreveld, M., Meijer, H., Speckmann, B., Verbeek, K.: On planar supports for hypergraphs. In: Eppstein, D., Gansner, E.R. (eds.) GD 2009. LNCS, vol. 5849, pp. 345–356. Springer, Heidelberg (2010), see also Tech. Rep. UU-CS-2009-035, Utrecht University (2009) Google Scholar
- 8.Chow, S.: Generating and Drawing Area-Proportional Euler and Venn Diagrams. PhD thesis, University of Victoria (2007)Google Scholar
- 9.Collins, C., Penn, G., Carpendale, S.: Bubble sets: Revealing set relations with isocontours over existing visualizations. IEEE TVCG 15(6), 1009–1016 (2009)Google Scholar
- 16.Kohonen, T.: Self-Organizing Maps, 3rd edn. Springer (2001)Google Scholar
- 20.S. Wagner and D. Wagner. Comparing Clusterings – An Overview. Tech. Rep. 2006-04, Department of Informatics, Universität Karlsruhe, 2007.Google Scholar