  We establish the existence and uniqueness of rational conformal maps of
minimal degree $n+1$ for opening up $n$ arcs. In earlier results, the degree
was exponential in $n$. We also discuss two related problems. (a) We establish
existence of rational functions of minimal degree with prescribed critical
values, and show that the number of (suitably normalized) rational functions is
given in terms of the Hurwitz numbers. (b) We consider the problem of finding
rational functions of minimal degree with prescribed critical points, where we
establish existence of solutions by considering certain polynomial equations,
and where the number of normalized solutions is bounded from above by a Catalan
number. We illustrate our results with two examples.
