  The article starts with some introductory material about resolution graphs of
normal surface singularities (definitions, topological/homological properties,
etc). We then discuss the case when the normal surface singularity is an N-fold
cyclic covering of a surface germ, branched along a curve given by the germ of
an analytic function f. We present non-trivial examples in order to show that
from the embedded resolution graph G of f in general it is not possible to
recover the resolution graph of the cyclic covering. The main results are the
construction of a ``universal covering graph'' from the topology of the germ f,
and the completely combinatorial construction of the resolution graph of cyclic
coverings from this universal graph of f and the integer N. For this we also
prove some purely graph-theoretical classification theorems of ``covering
graphs''. In the last part, we connect the properties of the universal covering
graph with the topological invariants of f, e.g. with the nilpotent part of its
algebraic monodromy.
