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 $\mathcal O(n^{r+1})$ of its hyperedges, despite its potentially exponential number of hyperedges. We also prove that there exist hypergraphs that need at least $\Omega(n^r)$ hyperedges to be represented, using a generalization of set orthogonality.

• 05C31 - Graph polynomials
• 05C62 - Graph representations (geometric and intersection representations, etc.)
• 05C65 - Hypergraphs
• 05C70 - Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)