On planar supports for hypergraphs

Authors:
Kevin Buchin;Marc van Kreveld;Henk Meijer;Bettina Speckmann;Kevin Verbeek
Affiliations:
Dep. of Mathematics and Computer Science, TU Eindhoven, The Netherlands;Dep. of Computer Science, Utrecht University, The Netherlands;Roosevelt Academy, Middelburg, The Netherlands;Dep. of Mathematics and Computer Science, TU Eindhoven, The Netherlands;Dep. of Mathematics and Computer Science, TU Eindhoven, The Netherlands
Venue:
GD'09 Proceedings of the 17th international conference on Graph Drawing
Year:
2009

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Abstract

A graph G is a support for a hypergraph $H = (V, \mathcal{S})$ if the vertices of G correspond to the vertices of H such that for each hyperedge $S_i \in \mathcal{S}$ the subgraph of G induced by Si is connected. G is a planar support if it is a support and planar. Johnson and Pollak [9] proved that it is NP-complete to decide if a given hypergraph has a planar support. In contrast, there are polynomial time algorithms to test whether a given hypergraph has a planar support that is a path, cycle, or tree. In this paper we present an algorithm which tests in polynomial time if a given hypergraph has a planar support that is a tree where the maximal degree of each vertex is bounded. Our algorithm is constructive and computes a support if it exists. Furthermore, we prove that it is already NP-hard to decide if a hypergraph has a 3-outerplanar support.