Simultaneous Embeddability of Two Partitions

Athenstädt, Jan Christoph and Hartmann, Tanja and Nöllenburg, Martin (2014) Simultaneous Embeddability of Two Partitions. In: Graph Drawing 22nd International Symposium, GD 2014, September 24-26, 2014 , pp. 64-75(Official URL:

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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.

Item Type: Conference Paper
Additional Information: 10.1007/978-3-662-45803-7_6
Classifications: G Algorithms and Complexity > G.490 Embeddings
G Algorithms and Complexity > G.999 Others
P Styles > P.180 Cluster
P Styles > P.420 Hyper
Z Theory > Z.500 Representations

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