  Class-S theories are four-dimensional N=2 supersymmetric field theories
constructed by the reduction of a (2,0) six-dimensional theory on a punctured
Riemann surface C (called the UV curve). A basic degeneration limit of the
surface C is when several punctures are brought close to each other. As this
happens, a long tube appears in C and a weakly coupled gauge group emerges.
When the corresponding gauge coupling is turned off, we are left with two
surfaces. This decoupling leads to a unique result. We explain how to fix the
gauge group that becomes weakly coupled and the resulting two theories, in
terms of the types of punctures on the surface C. Each of the resulting
theories is given in terms of a class-S theory (plus possibly additional free
hypermultiplets). Many other more complicated degeneration limits can be
described by repeatedly using this specification.
  For questions about weak coupling limits in such theories, it is useful to
consider the set of punctures that can appear at the end of a tube. These are
the punctures that are created in the resulting surfaces when the gauge
coupling is taken to be extremely weak. Not any puncture can be formed in a
decoupling process, and this set is described. Each such puncture can be found
diagrammatically in terms of the other punctures on C in a particular
decoupling limit. The relation of the gauge group to the symmetry associated
with this puncture is discussed.
  We discuss the relation of such degeneration limits of large N theories,
through the AdS/CFT correspondence, to decoupled field theories on AdS_5.
