A constructive characterisation of circuits in the simple -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 where the number of edges induced by any X ⊊ V is at most 2 | X | − 2 . Insisting on simplicity results in the Henneberg 2 operation being adequate only when the graph is sufficiently connected. Thus we introduce 3 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.