Hypergraph Representation via Axis-Aligned Point-Subspace Cover

TL;DR

This work introduces axis-aligned point-subspace covers as a geometric representation of $k$-partite $k$-uniform hypergraphs, defining $(d,\ell)$-hypergraphs with $k={\binom{d}{\ell}}$ and establishing a vertex-separability criterion that characterizes representability. It provides constructive, polynomial-time recognition algorithms for fixed dimension $d$ and prescribed vertex partition, first for point-line covers ($\ell=1$) and then generalized to arbitrary $\ell$, by reducing the problem to connected components of graphs $G_i$ and coordinating hyperedges and vertices via a coordinate-based mapping. The paper also extends the theory through a generalized separability concept, discusses complexity implications, and explores related directions such as NP-hardness of matching in $(3,1)$-hypergraphs and plane many-directions representations. Overall, the results offer a combinatorial framework for understanding when geometric axis-aligned representations exist and how to compute them, with potential impact on hypergraph optimization and geometric incidence problems.

Abstract

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.

Figures (8)

• Figure 1: A graph \ref{['fig:plane-ex']} and a hypergraph \ref{['fig:space-ex']} and their representations in $\mathbb{R}^2$ and $\mathbb{R}^3$, respectively.
• Figure 2: Two hypergraphs that cannot be represented by a set of points in $\mathbb{R}^3$ to be covered by lines. In \ref{['fig:negative-ex-2edges']}, any two lines corresponding to the shared vertices already define a unique point; in \ref{['fig:negative-ex-3edges']}, one of the points representing a hyperedge would have to lie on two skew lines, which is impossible.
• Figure 3: A hypergraph $G$ (on the left) that is edge-separable, but not vertex-separable (the vertices $v$ and $v'$ from $V_2$ are not separable). In a hypothetical representation of $G$ (on the right), the line $\ell^{v'}$ must simultaneously intersect $\ell^{u'}$ and $\ell^{w'}$ and therefore must be equal to $\ell^{v}$, which is impossible
• Figure 4: A hypergraph $G$, the graphs $G_1, G_2, G_3$, and the coordinates of the points and lines corresponding to the hyperedges and vertices. The dots placed instead of coordinates mean that those coordinates are free.
• Figure 5: A vertex \ref{['fig:3dm-vertex-gadget']} and an edge \ref{['fig:3dm-edge-gadget']} gadgets.

Theorems & Definitions (17)

• Definition 1
• Definition 2: Vertex separability
• Definition 3: Edge separability
• Lemma 1
• proof
• Definition 4
• Theorem 1
• proof
• Definition 5: Vertex separability
• Definition 6: Edge separability