Representation of Planar Hypergraphs by Contacts of Triangles

Many representation theorems extend from planar graphs to planar hypergraphs. The authors proved in [10] that every planar graph has a representation by contact of triangles. We prove here that this representation result extend to planar linear hypergraphs. Although the graph proof was simple and led to a linear time drawing algorithm, the extension for hypergraphs needs more work. The proof we give here relies on a combinatorial characterization of those hypergraphs which are representable by contact of segments in the plane, We propose some possible generalization directions and open problems, related to the order dimension of the incidence posets of hypergraphs