In this paper we study
normal surface singularities whose fundamental genus (:= the arithmetic genus of the
fundamental cycle) is equal or greater than 2. For those singularities, we define some
minimality conditions, and we study the relation between them. Further we define
some sequence of such singularities, which is analogous to elliptic sequence for elliptic
singularities. In the case of hypersurface singularities of Brieskorn type, we study
some properties of the sequences.