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:受图表和平面图的分裂分解启发,我们引入了美元分解的概念。我们关注的焦点是按序图的美元分解,我们证明它形成了具有若干属性的超大图。我们证明,这种超大图只能使用其顶层的$mathcal O(n ⁇ r+1})$来代表,尽管其顶层的顶层可能具有指数性数。我们还证明,存在需要至少$\Omega(n ⁇ r)$的超大图,使用一套正方位的概括来代表。