Generating punctured surface triangulations with degree at least 4

As a sequel of a previous paper by the authors, we present here a generating theorem for the family of triangulations of an arbitrary punctured surface with vertex degree ≥ 4. The method is based on a series of reversible operations termed reductions which lead to a minimal set of triangulations in such a way that all intermediate triangulations throughout the reduction process remain within the family. Besides contractible edges and octahedra, the reduction operations act on two new configurations near the surface boundary named quasi-octahedra and -components. It is also observed that another configuration called -component remains unaltered under any sequence of reduction operations. We show that one gets rid of -components by flipping appropriate edges.