  We compare Stein fillings and Milnor fibers for rational surface
singularities with reduced fundamental cycle. Deformation theory for this class
of singularities was studied by de Jong-van Straten in [dJvS98]; they
associated a germ of a singular plane curve to each singularity and described
Milnor fibers via deformations of this singular curve. We consider links of
surface singularities, equipped with their canonical contact structures, and
develop a symplectic analog of de Jong-van Straten's construction. Using planar
open books and Lefschetz fibrations, we describe all Stein fillings of the
links via certain arrangements of symplectic disks, related by a homotopy to
the plane curve germ of the singularity. As a consequence, we show that many
rational singularities in this class admit Stein fillings that are not strongly
diffeomorphic to any Milnor fibers. This contrasts with previously known cases,
such as simple and quotient surface singularities, where Milnor fibers are
known to give rise to all Stein fillings. On the other hand, we show that if
for a singularity with reduced fundamental cycle, the self-intersection of each
exceptional curve is at most -5 in the minimal resolution, then the link has a
unique Stein filling (given by a Milnor fiber).
