Symmetric Laman Theorems for the Groups $\mathcal{C}_2$ and $\mathcal{C}_s$

Abstract

For a bar and joint framework $(G,p)$ with point group $\mathcal{C}_3$ 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 $\mathcal{C}_2$ and $\mathcal{C}_s$ 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 $\mathcal{C}_2$ and $\mathcal{C}_s$ 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.