  Consider a fixed connected, finite graph $\Gamma$ and equip its vertices with
weights $p_i$ which are non-negative integers. We show that there is a finite
number of possibilities for the coefficients of the canonical cycle of a
numerically Gorenstein surface singularity having $\Gamma$ as the dual graph of
the minimal resolution, the weights $p_i$ of the vertices being the arithmetic
genera of the corresponding irreducible components. As a consequence we get
that if $\Gamma$ is not a cycle, then there is a finite number of possibilities
of self-intersection numbers which one can attach to the vertices which are of
valency $\geq 2$, such that one gets the dual graph of the minimal resolution
of a numerically Gorenstein surface singularity. Moreover, we describe
precisely the situations when there exists an infinite number of possibilities
for the self-intersections of the component corresponding to some fixed vertex
of $\Gamma$.
