Symmetric Laman theorems for the groups

For a bar and joint framework (G, p) with point group C3 which describes 3-fold rotational symmetry in the plane, it was recently shown in (Schulze, Discret. Comp. Geom. 44:946-972) that the standard Laman conditions, together with the condition derived in (Connelly et al., Int. J. Solids Struct. 46:762-773) that no vertices are fixed by the automorphism corresponding to the 3-fold rotation (geometrically, no vertices are placed on the center of rotation), are both necessary and sufficient for (G, p) to be isostatic, provided that its joints are positioned as generically as possible subject to the given symmetry constraints. In this paper we prove the analogous Laman-type conjectures for the groups C2 and Cs which are generated by a half-turn and a reflection in the plane, respectively. In addition, analogously to the results in (Schulze, Discret. Comp. Geom. 44:946-972), we also characterize symmetry generic isostatic graphs for the groups C2 and Cs in terms of inductive Henneberg-type constructions, as well as Crapo-type 3Tree2 partitions the full sweep of methods used for the simpler problem without symmetry.