Generating families of surface triangulations. The case of punctured surfaces with inner degree at least 4

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.