Rigid cylindrical frameworks with two coincident points

We develop a rigidity theory for graphs whose vertices are constrained to lie on a cylinder and in which two given vertices are coincident. We apply our result to show that the vertex splitting operation preserves the global rigidity of generic frameworks on the cylinder, whenever it satisfies the necessary condition that the deletion of the edge joining the split vertices preserves generic rigidity

Rigid Cylindrical Frameworks with Two Coincident Points

We develop a rigidity theory for graphs whose vertices are constrained to lie on a cylinder and in which two given vertices are coincident. We apply our result to show that the vertex splitting operation preserves the global rigidity of generic frameworks on the cylinder, whenever it satisfies the necessary condition that the deletion of the edge joining the split vertices preserves generic rigidity

Global rigidity of generic frameworks on the cylinder

We show that a generic framework (G,p) on the cylinder is globally rigid if and only if G is a complete graph on at most four vertices or G is both redundantly rigid and 2-connected. To prove the theorem we also derive a new recursive construction of circuits in the simple (2,2)-sparse matroid, and a characterisation of rigidity for generic frameworks on the cylinder when a single designated vertex is allowed to move off the cylinder

Rigidity of symmetric frameworks on the cylinder

A bar-joint framework $(G,p)$ is the combination of a finite simple graph $G=(V,E)$ and a placement $p:V\rightarrow \mathbb{R}^d$. The framework is rigid if the only edge-length preserving continuous deformations of the vertices arise from isometries of the space. This article combines two recent extensions of the generic theory of rigid and flexible graphs by considering symmetric frameworks in $\mathbb{R}^3$ restricted to move on a surface. In particular necessary combinatorial conditions are given for a symmetric framework on the cylinder to be isostatic (i.e. minimally infinitesimally rigid) under any finite point group symmetry. In every case when the symmetry group is cyclic, which we prove restricts the group to being inversion, half-turn or reflection symmetry, these conditions are then shown to be sufficient under suitable genericity assumptions, giving precise combinatorial descriptions of symmetric isostatic graphs in these contexts.Comment: 37 pages, 17 figure