Hypergraph Representation via Axis-Aligned Point-Subspace Cover
A k-hypergraph is a k-partite k-uniform hypergraph, that is, a hypergraph with a partition of vertices into k parts such that each hyperedge contains exactly one vertex of each part. We propose a new geometric representation of k-hypergraphs. Namely, given positive integers l, d, and k with l [less...
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| Published in: | Discrete Mathematics and Theoretical Computer Science Vol. 27; no. 2; pp. 1 - 15 |
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| Language: | English |
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01.08.2025
Discrete Mathematics & Theoretical Computer Science |
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| ISSN: | 1462-7264, 1365-8050 |
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| Abstract | A k-hypergraph is a k-partite k-uniform hypergraph, that is, a hypergraph with a partition of vertices into k parts such that each hyperedge contains exactly one vertex of each part. We propose a new geometric representation of k-hypergraphs. Namely, given positive integers l, d, and k with l [less than or equal to] d - 1 and [Please download the PDF to view the mathematical expression], any finite set P of points in [R.sup.d] represents a k-hypergraph [G.sub.p] as follows. Each point in P is covered by k many axis-aligned affine l-dimensional subspaces of [R.sup.d], which we call l-subspaces for brevity and which form the vertex set of [G.sub.p]. We interpret each point in P as a hyperedge of [G.sub.p] that contains each of the covering l-subspaces as a vertex. The class of (d, l)-hypergraphs is the class of k-hypergraphs that can be represented in this way. The resulting classes of hypergraphs are fairly rich, since every k-hypergraph is a (k, k - 1)-hypergraph. On the other hand, for l < d - 1, there exists a k-hypergraph which is not a (d, l)-hypergraph. In this paper we give a natural structural characterization of (d, l)-hypergraphs based on vertex cuts. This characterization leads to a polynomial-time recognition algorithm that decides for a given k-hypergraph G whether or not G is a (d, l)-hypergraph and that computes a representation of G if one exists. Here we assume that the dimension d is constant and that the partition of the vertex set of G is prescribed. Keywords: hypergraph, point-line cover, graph representation |
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| AbstractList | A k-hypergraph is a k-partite k-uniform hypergraph, that is, a hypergraph with a partition of vertices into k parts such that each hyperedge contains exactly one vertex of each part. We propose a new geometric representation of k-hypergraphs. Namely, given positive integers l, d, and k with l [less than or equal to] d - 1 and [Please download the PDF to view the mathematical expression], any finite set P of points in [R.sup.d] represents a k-hypergraph [G.sub.p] as follows. Each point in P is covered by k many axis-aligned affine l-dimensional subspaces of [R.sup.d], which we call l-subspaces for brevity and which form the vertex set of [G.sub.p]. We interpret each point in P as a hyperedge of [G.sub.p] that contains each of the covering l-subspaces as a vertex. The class of (d, l)-hypergraphs is the class of k-hypergraphs that can be represented in this way. The resulting classes of hypergraphs are fairly rich, since every k-hypergraph is a (k, k - 1)-hypergraph. On the other hand, for l < d - 1, there exists a k-hypergraph which is not a (d, l)-hypergraph. 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. A k-hypergraph is a k-partite k-uniform hypergraph, that is, a hypergraph with a partition of vertices into k parts such that each hyperedge contains exactly one vertex of each part. We propose a new geometric representation of khypergraphs. Namely, given positive integers l, d, and k with l ≤ d - 1 and k = d l , any finite set P of points in Rd represents a k-hypergraph GP as follows. Each point in P is covered by k many axis-aligned affine l-dimensional subspaces of Rd, which we call l-subspaces for brevity and which form the vertex set of GP . We interpret each point in P as a hyperedge of GP that contains each of the covering l-subspaces as a vertex. The class of (d, l)-hypergraphs is the class of k-hypergraphs that can be represented in this way. The resulting classes of hypergraphs are fairly rich, since every k-hypergraph is a (k, k - 1)-hypergraph. On the other hand, for l < d - 1, there exists a k-hypergraph which is not a (d, l)-hypergraph. In this paper we give a natural structural characterization of (d, l)-hypergraphs based on vertex cuts. This characterization leads to a polynomial-time recognition algorithm that decides for a given k-hypergraph G whether or not G is a (d, l)-hypergraph and that computes a representation of G if one exists. Here we assume that the dimension d is constant and that the partition of the vertex set of G is prescribed. A k-hypergraph is a k-partite k-uniform hypergraph, that is, a hypergraph with a partition of vertices into k parts such that each hyperedge contains exactly one vertex of each part. We propose a new geometric representation of k-hypergraphs. Namely, given positive integers l, d, and k with l [less than or equal to] d - 1 and [Please download the PDF to view the mathematical expression], any finite set P of points in [R.sup.d] represents a k-hypergraph [G.sub.p] as follows. Each point in P is covered by k many axis-aligned affine l-dimensional subspaces of [R.sup.d], which we call l-subspaces for brevity and which form the vertex set of [G.sub.p]. We interpret each point in P as a hyperedge of [G.sub.p] that contains each of the covering l-subspaces as a vertex. The class of (d, l)-hypergraphs is the class of k-hypergraphs that can be represented in this way. The resulting classes of hypergraphs are fairly rich, since every k-hypergraph is a (k, k - 1)-hypergraph. On the other hand, for l < d - 1, there exists a k-hypergraph which is not a (d, l)-hypergraph. In this paper we give a natural structural characterization of (d, l)-hypergraphs based on vertex cuts. This characterization leads to a polynomial-time recognition algorithm that decides for a given k-hypergraph G whether or not G is a (d, l)-hypergraph and that computes a representation of G if one exists. Here we assume that the dimension d is constant and that the partition of the vertex set of G is prescribed. Keywords: hypergraph, point-line cover, graph representation |
| Audience | Academic |
| Author | Firman, Oksana Spoerhase, Joachim |
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| Snippet | A k-hypergraph is a k-partite k-uniform hypergraph, that is, a hypergraph with a partition of vertices into k parts such that each hyperedge contains exactly... 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... |
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| SubjectTerms | Algorithms Apexes computer science - discrete mathematics Graph theory Graphs Mathematical research mathematics - combinatorics Polynomials Representations Structural analysis Subspaces Vertex sets |
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| Title | Hypergraph Representation via Axis-Aligned Point-Subspace Cover |
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