  In this article, we generalize the results discussed in [arXiv:1004.3762] by
introducing a genus to generic fibers of Lefschetz fibrations. That is, we give
families of relations in the mapping class groups of genus-1 surfaces with
boundaries that represent rational homology disk smoothings of weighted
homogeneous surface singularities whose resolution graphs are $3$-legged with a
bad central vertex.
