Tinkertoys for Gaiotto duality

We describe a procedure for classifying N = 2 superconformal theories of the type introduced by Davide Gaiotto. Any curve, C, on which the 6D A(N-1) SCFT is compactified, can be decomposed into 3-punctured spheres, connected by cylinders. We classify the spheres, and the cylinders that connect them. The classification is carried out explicitly, up through N = 5, and for several families of SCFTs for arbitrary N. These lead to a wealth of new S-dualities between Lagrangian and non-Lagrangian N = 2 SCFTs.