Hypergraph Representation via Axis-Aligned Point-Subspace Cover
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We propose a new representation of k-partite, k-uniform hypergraphs (i.e. 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) by a finite set P of points in ℝ^d and a parameter ℓ≤ d-1. Each point in P is covered by k=dℓ many axis-aligned affine ℓ-dimensional subspaces of ℝ^d, which we call ℓ-subspaces for brevity. We interpret each point in P as a hyperedge 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, where k=dℓ. The resulting classes of hypergraphs are fairly rich: Every k-hypergraph is a (k,k-1)-hypergraph. On the other hand, (d,ℓ)-hypergraphs form a proper subclass of the class of all dℓ-hypergraphs for ℓ<d-1. 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 dℓ-hypergraph whether or not it is a (d,ℓ)-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.
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