On Planar Supports for Hypergraphs
نویسندگان
چکیده
A graph G is a support for a hypergraph H = (V,S) if the vertices of G correspond to the vertices of H such that for each hyperedge Si ∈ 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 [11] proved that it is NPcomplete to decide if a given hypergraph has a planar support. In contrast, there are lienar 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 efficient 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 2-outerplanar support. Submitted: January 2010 Reviewed: January 2011 Revised: June 2011 Accepted: August 2011 Final: September 2011 Published: September 2011 Article type: Regular Paper Communicated by: M. Kaufmann E-mail addresses: [email protected] (Kevin Buchin) [email protected] (Marc van Kreveld) [email protected] (Henk Meijer) [email protected] (Bettina Speckmann) [email protected] (Kevin Verbeek) 534 Buchin et al. On Planar Supports for Hypergraphs
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