We propose a new representation of $k$-partite, $k$-uniform hypergraphs, that is, a hypergraph with a partition of vertices into $k$ parts such that each hyperedge contains exactly one vertex of each type; we call them $k$-hypergraphs for short. Given positive integers $\ell, d$, and $k$ with $\ell\leq d-1$ and $k={d\choose\ell}$, any finite set $P$ of points in $\mathbb{R}^d$ represents a $k$-hypergraph $G_P$ as follows. Each point in $P$ is covered by $k$ many axis-aligned affine $\ell$-dimensional subspaces of $\mathbb{R}^d$, which we call $\ell$-subspaces for brevity and which form the vertex set of $G_P$. We interpret each point in $P$ as a hyperedge of $G_P$ that contains each of the covering $\ell$-subspaces as a vertex. The class of \emph{$(d,\ell)$-hypergraphs} is the class of $k$-hypergraphs that can be represented in this way. The resulting classes of hypergraphs are fairly rich: Every $k$-hypergraph is a $(k,k-1)$-hypergraph. On the other hand, $(d,\ell)$-hypergraphs form a proper subclass of the class of all $k$-hypergraphs for $\ell<d-1$. In this paper we give a natural structural characterization of $(d,\ell)$-hypergraphs based on vertex cuts. This characterization leads to a poly\-nomial-time recognition algorithm that decides for a given $k$-hypergraph whether or not it is a $(d,\ell)$-hypergraph and that computes a representation if existing. We assume that the dimension $d$ is constant and that the partitioning of the vertex set is prescribed.

翻译：我们提出了一种关于 $k$ 部、$k$ 一致超图的新表示方法，即顶点划分为 $k$ 个部分且每条超边恰好包含每类顶点各一个的超图；为简便起见，我们将其称为 $k$-超图。给定正整数 $\ell, d$ 和 $k$，满足 $\ell\leq d-1$ 且 $k={d\choose\ell}$，$\mathbb{R}^d$ 中任意有限点集 $P$ 均可按如下方式表示一个 $k$-超图 $G_P$：$P$ 中的每个点被 $\mathbb{R}^d$ 中 $k$ 个轴向对齐的仿射 $\ell$ 维子空间覆盖（为简洁起见，我们称其为 $\ell$-子空间），这些子空间构成 $G_P$ 的顶点集。我们将 $P$ 中的每个点解释为 $G_P$ 的一条超边，该超边包含所有覆盖该点的 $\ell$-子空间作为顶点。\emph{$(d,\ell)$-超图} 类即为能以此方式表示的所有 $k$-超图构成的类。由此产生的超图类相当丰富：每个 $k$-超图都是 $(k,k-1)$-超图。另一方面，当 $\ell<d-1$ 时，$(d,\ell)$-超图构成所有 $k$-超图类的真子类。本文基于顶点割给出了 $(d,\ell)$-超图的一个自然结构刻画。该刻画引出了一个多项式时间识别算法，该算法判定给定 $k$-超图是否为 $(d,\ell)$-超图，并在存在表示时计算出一个表示。我们假设维度 $d$ 为常数，且顶点集的划分是预先给定的。