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：受图的拆分分解和秩宽度的启发，我们引入了$r$-分割的概念。我们聚焦于$n$阶图的$r$-分割族，并证明该族构成一个具有若干性质的超图。我们证明，尽管此类超图可能包含指数级数量的超边，但仅需使用$\mathcal O(n^{r+1})$个超边即可表示它们。我们还利用集合正交性的推广证明，存在至少需要$\Omega(n^r)$个超边才能表示的超图。