Natural realizations of sparsity matroids

TL;DR

This paper introduces a natural linear representation theorem for (k,l)-sparse hypergraphs, connecting hypergraph sparsity conditions with matroid theory, motivated by rigidity problems.

Contribution

It provides a new, natural linear representation for (k,l)-sparse hypergraphs that reflects their incidence structure, advancing understanding in rigidity theory.

Findings

Linear representability of (k,l)-sparse hypergraphs

Representation captures hypergraph incidence structure

Application to rigidity theory

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'\le kn'-l. For integers k and l satisfying 0\le l\le 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.