René van Bevern, Iyad Kanj, Christian Komusiewicz, Rolf Niedermeier, and Manuel Sorge. Twins in subdivision drawings of hypergraphs. In Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD'16), Athens, Greece, volume 9801 of Lecture Notes in Computer Science, pages 67–80. Springer, 2016.

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 draw- ings 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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