Hypergraphs with Polynomial Representation: Introducing r-splits
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Inspired by the split decomposition of graphs and rank-width, we introduce the notion of r-splits. We focus on the family of r-splits of a graph of order n, and we prove that it forms a hypergraph with several properties. We prove that such hypergraphs can be represented using only 𝒪(n^r+1) of its hyperedges, despite its potentially exponential number of hyperedges. We also prove that there exist hypergraphs that need at least Ω(n^r) hyperedges to be represented, using a generalization of set orthogonality.
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