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 e 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  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.
|Title of host publication||Graph Drawing (17th International Symposium, GD'09, Chicago, IL, USA, September 22-25, 2009. Revised Papers)|
|Editors||D. Eppstein, E.R. Gansner|
|Place of Publication||Berlin|
|Publication status||Published - 2010|
|Name||Lecture Notes in Computer Science|