Hypergraphs with Polynomial Representation: Introducing r-splits

Inspired by the split decomposition of graphs and rank-width, we introduce the notion of r-splits. We focus on the family of r-splits of a graph of order n, and we prove that it forms a hypergraph with several properties. We prove that such hypergraphs can be represented using only O (nr+1) of its hyperedges, despite its potentially exponential number of hyperedges. We also prove that there exist hypergraphs that need at least omega(nr) hyperedges to be represented, using a generalization of set orthogonality.