  We present two versions of a method for generating all triangulations of any
punctured surface in each of these two families: (1) triangulations with inner
vertices of degree at least 4 and boundary vertices of degree at least 3 and
(2) triangulations with all vertices of degree at least 4. The method is based
on a series of reversible operations, termed reductions, which lead to a
minimal set of triangulations in each family. Throughout the process the
triangulations remain within the corresponding family. Moreover, for the family
(1) these operations reduce to the well-known edge contractions and removals of
octahedra. The main results are proved by an exhaustive analysis of all
possible local configurations which admit a reduction.
