Congruence condition for resolution graphs

Let $(X,o)$ be a ${\mathbb Q}$-Gorenstein surface singularity with ${\mathbb Q}$-homology sphere link, and let its resolution graph be equipped with the associated discriminant-group action on the splice-diagram complete intersection. Congruence condition. The resolution graph satisfies the congruence conditions required for the discriminant group to act freely away from the origin on that complete intersection. These conditions are additional to the semigroup condition and depend on the resolution graph, not only on its splice diagram; the paper verifies them in several classes but leaves the general assertion open.