Sparse hypergraphs with applications in combinatorial rigidity

A hypergraph H = (V,E) is called (1, k)-sparse, for some integer k, if each subset X ⊆ V with |X| ≥ k spans at most |X| − k hyperedges. If we also have |E| = |V | − k then H is (1, k)-tight. Hypergraphs of this kind occur in several problems of combinatorial rigidity, where the goal is to analyse the generic rigidity properties of point sets equipped with geometric constraints involving subsets of points. Motivated by this connection we develop a new inductive construction of 4-regular (1, 3)-tight hypergraphs and use it to deduce a Lamantype combinatorial characterization of generically minimally rigid projective frameworks on the projective line. Hypergraphs with the same sparsity parameter show up in some key results of scene analysis, due to Whiteley, as well as in affine rigidity, introduced by Gortler et al. Thus our result implies a Henneberg-type inductive construction of generically minimally rigid affine frameworks in the plane. Based on the rank function of the corresponding count matroid on the edge set of H we also obtain purely combinatorial proofs for some results on generically affinely rigid hypergraphs.