  In \cite{NNM} the author with A. N\'emethi computed the multiplicity of
generic surface singularities, the formula is purely topological computable
from the resolution graph of the surface singularity. In the present paper we
extend the results partly to the relative case, when there is a pair of
resolution graphs $\mathcal{T}_1 \subset \mathcal{T}$, a fixed singularity
$\tX_1$ with resolution graph $\mathcal{T}_1$, a relatively generic singularity
$\tX$ corresponding to the subsingularity $\tX_1$ with resolution graph
$\mathcal{T}$.
  We determine the base points of the natural line bundles on $\tX$ from the
resoultion graph $\mathcal{T}$ and the analytic invariants of the
subsingularity $\tX_1$. For each base point $p$ we determine a lower bound for
the number $t_p$ such that $p$ is $t_p$-simple and we compute from it a lower
bound of the multiplicity of $\tX$, which is sharp in all known cases.
