Bracing frameworks consisting of parallelograms

TL;DR

This paper extends the study of rigidity in braced frameworks from grids and rhombic carpets to more general parallelogram-based structures, establishing a connection between rigidity and graph connectivity.

Contribution

It generalizes the concept of bracing frameworks to include any graph with parallelogram cycles and proves the equivalence between rigidity, edge coloring, and bracing graph connectivity.

Findings

Rigidity is equivalent to the non-existence of a special edge coloring.

Rigidity corresponds to the connectivity of the bracing graph.

The framework applies to a broader class of parallelogram-based structures.

Abstract

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.