Equivalent Separating Curves on Surfaces and Its Applications to Haken Spheres

1 Introduction Let A be a collection of n pairwise disjoint simple closed curves on an orientable closed surface F of genus n ≥2. We say that A is a complete system of F if the surface obtained by cutting F along A is a 2n-punctured sphere. Let A1, A2 be two non-empty subsets of A. We say that (A1,A2) is a partition of A if A1 ∩A2 = (?) and A1 ∪A2 = A. Let (A1, A2) be a partition of A on F, and C a simple closed curve on F. We say that C is separating with respect to (A1, A2) if it is disjoint from A and it cuts F into two pieces F1,F2 with A1(?) F1,A2 (?) F2.