A Constructive Characterisation of Circuits in the Simple (2,2)-sparsity Matroid

We provide a constructive characterisation of circuits in the simple (2, 2)sparsity matroid. A circuit is a simple graph G = (V,E) with |E| = 2|V | − 1 and the number of edges induced by any X ( V is at most 2|X| − 2. Insisting on simplicity results in the Henneberg operation being enough only when the graph is sufficiently connected. Thus we introduce 3 different sum 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.