N-Flips in Triangulations with Two Odd Degree Vertices

Abstract

In this paper, we prove that any two triangulations G and $$G'$$ on the sphere with exactly two odd degree vertices can be transformed into each other by two local transformations, called an N-flip and a $$P_2$$-flip, preserving the parity of degree of each vertex, if $$|V(G)|=|V(G')|$$. This is an analogy of the same result for triangulations with each vertex even degree [7], but we prove such a fact does not hold for triangulations with at least four odd degree vertices.