Keywords: bar joint framework, cubic grid, rigidity, scaffolding, directed graph, computational complexity.

Bar and joint frameworks present models of engineering structures. The purpose is to find an efficient algorithm for deciding infinitesimal rigidity in differently braced three-dimensional. Using the bar-joint structure's symmetry to determine the rigidity is a problem of long-standing interest in kinematics, statics, and optimization. The algorithm has applications in robotics as an actuator-controlled mechanism and in material science as meta-materials and reconfigurable materials. The bar and joint framework have served as valuable models of the structure of metals, crystal states of matter, building science, and biological systems. Scaffolding, as repetitive objects, are helpful as preliminary structures of design. Applying some further bracing elements such as Cable, Strut, or Rod (bracing bar) the Scaffolding will be rigid. The given models describe the rigidity of the differently braced scaffolding frameworks and produce a graph theoretical characterization that provides an efficiently solvable graph or directed graph as the original structure.

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