Abstract
Combinatorial characterisations of minimal rigidity are obtained for symmetric $2$-dimensional bar-joint frameworks with either $\ell^1$ or $\ell^\infty$ distance constraints. The characterisations are expressed in terms of symmetric tree packings and the number of edges fixed by the symmetry operations. The proof uses new Henneberg-type inductive construction schemes.
Highlights
A fundamental problem in geometric rigidity theory is to find combinatorial characterisations of graphs which form rigid bar-joint frameworks for all generic realisations of the vertices in a given space
Generic rigidity has been characterised in terms of tree packings for body-bar, body-hinge and molecular frameworks
If G is expressible as a union of two edge-disjoint spanning trees, both of which are Z2-symmetric with respect to θ, and no edge of G is fixed by s there exists a construction chain, W5 = G1 → G2 → · · · → Gn = G, such that for each k = 1, 2, . . . , n − 1, (i) Gk → Gk+1 is an allowable (Z2, θ) graph extension, and, (ii) Gk is expressible as a union of two edge-disjoint spanning trees, both of which are Z2-symmetric with respect to θ, and no edge of Gk is fixed by s
Summary
A fundamental problem in geometric rigidity theory is to find combinatorial characterisations of graphs which form rigid bar-joint frameworks for all generic realisations of the vertices in a given space. The graphs H1 and H2 are edge-disjoint Z2-symmetric spanning trees in H and it follows that (H, θ) is an admissible pair.
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