  We classify the resolution graphs of weighted homogeneous surface
singularities which admit rational homology disk smoothings. The nonexistence
of rational homology disk smoothings is shown by symplectic geometric methods,
while the existence is verified via smoothings of negative weights. In
particular, it is shown that a (negative definite) starshaped plumbing tree
gives rise to a weighted homogeneous singularity admitting a rational homology
disk smoothing if and only if the Milnor fillable contact structure of the link
admits a weak symplectic rational homology disk filling.
