  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.
