A finiteness theorem for dual graphs of surface singularities

Consider a fixed connected, finite graph $\mathrm{\Gamma }$ 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 $\mathrm{\Gamma }$ 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 $\mathrm{\Gamma }$ 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 $\ge 2$, 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 $\mathrm{\Gamma }$.