Hypergraphs with Polynomial Representation: Introducing $r$-splits
François Pitois, Mohammed Haddad, Hamida Seba, Olivier Togni
TL;DR
This work introduces $r$-splits as a rank-$$r$ generalization of split decomposition, framing the collection of all $r$-splits of a graph on $n$ vertices as a hypergraph $H_r(G)$. Leveraging a closure framework with ${<\cdot\!>_r}$ and a generalized notion of orthogonality, the authors prove that $H_r(G)$ admits a compact representation with $\mathcal{O}(n^{r+1})$ hyperedges via a polynomial decomposition into essential hyperedges. They define $r$-orthogonality and $r$-cross-free hypergraphs to analyze structural sparsity, showing upper bounds for cross-free closures and constructing families achieving $\Omega(n^r)$ hyperedges for representation, thereby establishing a gap between lower and upper bounds. The results lay groundwork for broader decompositions akin to rank-width, while highlighting potential for efficient representations and highlighting fundamental limits via lower-bound constructions. Overall, the paper contributes to understanding how rank-based cuts induce hypergraph representations and how closure and orthogonality notions control representational complexity in this setting.
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 $Ω(n^r)$ hyperedges to be represented, using a generalization of set orthogonality.