Toroidalization of generating sequences in dimension two function fields

Let k be an algebraically closed field of characteristic 0, and let K*/K be a finite extension of algebraic function fields of transcendence degree 2 over k. Let ν* be a k-valuation of K* with valuation ring V*, and let ν be the restriction of ν* to K. Suppose that R→S is an extension of algebraic regular local rings with quotient fields K and K* respectively, such that V* dominates S and S dominates R. We prove that there exist sequences of quadratic transforms and along ν* such that dominates and the map between generating sequences of ν and ν* has a toroidal structure. Our result extends the Strong Monomialization theorem of Cutkosky and Piltant.