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.

翻译：受图的split分解和秩宽度的启发，我们引入了$r$-分割的概念。针对阶数为$n$的图的$r$-分割族，我们证明其构成一个具有多种性质的hypergraph。我们证明，尽管此类hypergraph的超边数量可能呈指数级增长，但仅需$\mathcal O(n^{r+1})$条超边即可表示。此外，利用集合正交性的推广，我们证明存在至少需要$\Omega(n^r)$条超边才能表示的hypergraph。