On Planar Supports for Hypergraphs

Abstract

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 S_i ∈ S the subgraph of G induced by S_i 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 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.