Hypergraph Representation via Axis-Aligned Point-Subspace Cover - Firman, Oksana; Spoerhase, Joachim (Discrete Mathematics and Theoretical Computer Science, 2025)
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