N -Flips in Triangulations with Two Odd Degree Vertices

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.