Rational Contractions of Fiber Type on ${\bar{M}}_{0,6}$

We identify a set of initial rational contractions of fiber type on ${\bar{M}}_{0,6}$. Our proof uses a new algorithm we develop for verifying descriptions of the cone of effective divisors on varieties without elementary rational contractions of fiber type. On Mori dream spaces, our algorithm identifies faces of the cone of nef curves associated to initial rational contractions of fiber type.