Sparse Hypergraphs and Pebble Game Algorithms [article]

Ileana Streinu, Louis Theran
2007 arXiv   pre-print
A hypergraph G=(V,E) is (k,ℓ)-sparse if no subset V'⊂ V spans more than k|V'|-ℓ hyperedges. We characterize (k,ℓ)-sparse hypergraphs in terms of graph theoretic, matroidal and algorithmic properties. We extend several well-known theorems of Haas, Lovász, Nash-Williams, Tutte, and White and Whiteley, linking arboricity of graphs to certain counts on the number of edges. We also address the problem of finding lower-dimensional representations of sparse hypergraphs, and identify a critical
more » ... r in terms of the sparsity parameters k and ℓ. Our constructions extend the pebble games of Lee and Streinu from graphs to hypergraphs.
arXiv:math/0703921v1 fatcat:iwctiihw6rhxxmhzta7ansb46e