In arXiv:1809.03737 and arXiv:1809.03744 the authors investigated invariants of generic analytic structures on surface singularities and determined some of them like the geometric genus of generic surface singularities or $h^1$ of generic line bundles with a given Chern class on an arbitrary surface singularity. There are however other invariants of generic surface singularities, like multiplicity or embedded dimension we wish to compute in following manuscripts in the case of generic singularities... In the present article we work out a relative setup of generic structures on surface singularities, where we fix a given analytic type or line bundle on a smaller subgraph or more generally on a smaller cycle and we choose a relatively generic line bundle or analytic type on the large cycle and we wish to compute it's invariants, like geometric genus or $h^1$. The formulas which give the answers to these questions are intresting on their own, however the real power of these results, that they give possibility for inductive proofs of problems regarding generic surface singularities. (read more)