  In this article we give a construction of the resolution graphs of
hypersurface surface singularities (X_k,0) given by generalized Iomdin series.
All these resolution graphs are coordinated by an ``universal bi-colored
graph'' which is associated with the ICIS determining the Iomdin series. The
definition of this new graph is rather involved, and in concrete examples it is
difficult to compute. Nevertheless, we present a large number of examples. This
is very helpful in the exemplification of its properties as well. We then
present a construction of the resolution graphs of the surface singularities
above which uses the "universal bi-coloured graph" and the integer k. This is
formulated in a purely combinatorial algorithm. The result is a highly
non-trivial generalization of the case of cyclic coverings of smooth surfaces.
