Global rigidity of (quasi-)injective frameworks on the line

Garamvölgyi, Dániel

doi:10.1016/j.disc.2021.112687

Cited by 2 publications

(1 citation statement)

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“…This result was generalized to periodic [5] and rotational symmetry preserving flexes [6,7]. Checking whether a general finite graph has a NAC-coloring is NP-complete [9]. Similarly to [11] we need a specific type of NAC-colorings in this paper.…”

Section: Theorem 110 ([7 12]) a Connected Non-trivial Graph Allows A ...mentioning

confidence: 99%

Bracing frameworks consisting of parallelograms

Grasegger

Legerský

2022

Art Discrete Appl. Math.

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A rectangle in the plane can be continuously deformed preserving its edge lengths, but adding a diagonal brace prevents such a deformation. Bolker and Crapo characterized combinatorially which choices of braces make a grid of squares infinitesimally rigid using a bracing graph: a bipartite graph whose vertices are the columns and rows of the grid, and a row and column are adjacent if and only if they meet at a braced square. Duarte and Francis generalized the notion of the bracing graph to rhombic carpets, proved that the connectivity of the bracing graph implies rigidity and stated the other implication without proof. Nagy Kem gives the equivalence in the infinitesimal setting. We consider continuous deformations of braced frameworks consisting of a graph from a more general class and its placement in the plane such that every 4-cycle forms a parallelogram. We show that rigidity of such a braced framework is equivalent to the non-existence of a special edge coloring, which is in turn equivalent to the corresponding bracing graph being connected.

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