A constructive characterisation of circuits in the simple (2,2)(2,2)-sparsity matroid

We provide a constructive characterisation of circuits in the simple (2,2)(2,2)-sparsity matroid. A circuit is a simple graph G=(V,E)G=(V,E) with |E|=2|V|−1|E|=2|V|−1 where the number of edges induced by any X⊊VX⊊V is at most 2|X|−22|X|−2. Insisting on simplicity results in the Henneberg 2 operation being adequate only when the graph is sufficiently connected. Thus we introduce 33 different join operations to complete the characterisation. Extensions are discussed to when the sparsity matroid is connected and this is applied to the theory of frameworks on surfaces, to provide a conjectured characterisation of when frameworks on an infinite circular cylinder are generically globally rigid.