Using the picture deformation technique of De Jong-Van Straten we show
that no singularity whose resolution graph has 3 or 4 large nodes, i.e.,
nodes satisfying $d(v)+e(v)\leq -2$, has a QHD smoothing. This is achieved
by providing a general reduction algorithm for graphs with QHD
smoothings, and enumeration. New examples and families are presented,
which admit a combinatorial QHD smoothing, i.e. the incidence relations
for a sandwich presentation can be satisfied. We also give a new proof
of the Bhupal-Stipsicz theorem on the classification of weighted
homogeneous singularities admitting QHD smoothings with this method by
using cusp singularities.