The separating gonality of a separating real curve
  A smooth real curve is called separating in case the complement of the real locus inside the complex locus is disconnected. This is the case if there exists a morphism to the projective line whose inverse image of the real locus of the projective line is the real locus of the curve. Such morphism is called a separating morphism. The minimal degree of a separating morphism is called the separating gonality. The separating gonality cannot be less than the number s of the connected components of the real locus of the curve. A theorem of Ahlfors implies this separating gonality is at most the g+1 with g the genus of the curve. A better upper bound depending on s is proved by Gabard. In this paper we prove that there are no more restrictions on the values of the separating gonality. 
Marc Coppens
math.AG