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arXiv · 1511.09389

The role of twins in computing planar supports of hypergraphs

René van Bevern·Iyad A. Kanj·Christian Komusiewicz·Rolf Niedermeier·Manuel Sorge
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Abstract

A support or realization of a hypergraph $H$ is a graph $G$ on the same vertex as $H$ such that for each hyperedge of $H$ it holds that its vertices induce a connected subgraph of $G$. The NP-hard problem of finding a planar support has applications in hypergraph drawing and network design. Previous algorithms for the problem assume that twins -- pairs of vertices that are in precisely the same hyperedges -- can safely be removed from the input hypergraph. We prove that this assumption is generally wrong, yet that the number of twins necessary for a hypergraph to have a planar support only depends on its number of hyperedges. We give an explicit upper bound on the number of twins necessary for a hypergraph with $m$ hyperedges to have an $r$-outerplanar support, which depends only on $r$ and $m$. Since all additional twins can be safely removed, we obtain a linear-time algorithm for computing $r$-outerplanar supports for hypergraphs with $m$ hyperedges if $m$ and $r$ are constant; in other words, the problem is fixed-parameter linear-time solvable with respect to the parameters $m$ and $r$.

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René van Bevern, Iyad A. Kanj, Christian Komusiewicz, Rolf Niedermeier, Manuel Sorge. 2015-11-15. The role of twins in computing planar supports of hypergraphs. https://arxiv.org/abs/1511.09389

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