  In this paper, we classify Hamiltonian $S^1$-actions on compact, four
dimensional symplectic orbifolds that have isolated singular points with cyclic
orbifold structure groups, thus extending the classification due to Karshon to
the orbifold setting. To such a space, we associated a combinatorial invariant,
a labeled multigraph, that determines the isomorphism type of the space.
Moreover, we show that any such space can be obtained by applying finitely many
equivariant weighted blow-ups to a minimal space, i.e., one on which no
equivariant weighted blow-down can be applied.
