Natural realizations of sparsity matroids

Abstract

A hypergraph G with n vertices and m hyperedges with d endpoints each is (k; ℓ)- sparse if for all sub-hypergraphs G′ on n′ vertices and m′ edges, m′ < kn′ - ℓ. For integers k and ℓ satisfying 0 ≤ ℓ ≤ dk - 1, this is known to be a linearly representable matroidal family. Motivated by problems in rigidity theory, we give a new linear representation theorem for the (k; ℓ)-sparse hypergraphs that is natural; i.e., the representing matrix captures the vertex-edge incidence structure of the underlying hypergraph G. Copyright © 2011 DMFA Slovenije.