1. Hypergraphs with Polynomial Representation: Introducing $r$-splits
- Author
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Haddad, Mohammed, Pitois, François, Seba, Hamida, and Togni, Olivier
- Subjects
FOS: Computer and information sciences ,Discrete Mathematics (cs.DM) ,FOS: Mathematics ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,Computer Science - Discrete Mathematics - Abstract
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 $\mathcal O(n^{r+1})$ of its hyperedges, despite its potentially exponential number of hyperedges. We also prove that there exist hypergraphs that need at least $\Omega(n^r)$ hyperedges to be represented, using a generalization of set orthogonality.
- Published
- 2022
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