Twins in Subdivision Drawings of Hypergraphs
Visualizing hypergraphs, systems of subsets of some universe, has continuously attracted research interest in the last decades. We study a natural kind of hypergraph visualization called subdivision drawings. Dinkla et al. [Comput. Graph. Forum ’12] claim
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Novosibirsk State University, Novosibirsk, Russian Federation [email protected] Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russian Federation 3 DePaul University, Chicago, USA [email protected] 4 Friedrich-Schiller-Universit¨ at Jena, Jena, Germany [email protected] 5 TU Berlin, Berlin, Germany {rolf.niedermeier,manuel.sorge}@tu-berlin.de
Abstract. Visualizing hypergraphs, systems of subsets of some universe, has continuously attracted research interest in the last decades. We study a natural kind of hypergraph visualization called subdivision drawings. Dinkla et al. [Comput. Graph. Forum ’12] claimed that only few hypergraphs have a subdivision drawing. However, this statement seems to be based on the assumption (also used in previous work) that the input hypergraph does not contain twins, pairs of vertices which are in precisely the same hyperedges (subsets of the universe). We show that such vertices may be necessary for a hypergraph to admit a subdivision drawing. As a counterpart, we show that the number of such “necessary twins” is upper-bounded by a function of the number m of hyperedges and a further parameter r of the desired drawing related to its number of layers. This leads to a linear-time algorithm for determining such subdivision drawings if m and r are constant; in other words, the problem is linear-time fixed-parameter tractable with respect to the parameters m and r.
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Introduction
Hypergraph drawings are useful as visual aid in diverse applications [1], among them electronic circuit design [11] and relational databases [2,18]. There are several methods for embedding hypergraphs in the plane. The combinatorial problem studied in this work stems from obtaining subdivision drawings [14,15]. Significant parts of the work were done while all authors were with TU Berlin and while RvB, IK, and MS were supported by the DFG, project NI 369/12. R. van Bevern—Supported by grant 16-31-60007 mol a dk of the Russian Foundation for Basic Research. C. Komusiewicz—Supported by the DFG, project KO 3669/4-1. c Springer International Publishing AG 2016 Y. Hu and M. N¨ ollenburg (Eds.): GD 2016, LNCS 9801, pp. 67–80, 2016. DOI: 10.1007/978-3-319-50106-2 6
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Herein, given a hypergraph H, we divide the plane into closed regions that one-toone correspond to the vertices of H in such a way that, for each hyperedge F , the union of the regions corresponding to the vertices in F is connected. Subdivision drawings have also been called vertex-based Venn diagrams [14]. Figure 1 shows an example for such a drawing.
Fig. 1. Two drawings of the same hypergraph. On the left, we see a drawing in the subset standard in which the vertices (white circles) are enclosed by curves that correspond to hyperedges. On the right, we see a subdivision drawing in which we assign vertices to regions (enclosed by black lines) and we color these regions with colors that one-to-one correspond to the hyperedges; for each hyperedge, the union of th
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