Applications of combinatorics to statics—rigidity of grids Original Research Article

The infinitesimal rigidity (or briefly rigidity) of a bar-and-joint framework (in any dimension) can be formulated as a rank condition of the so-called rigidity matrix. If there are n joints in the framework then the size of this matrix is O(n), so the time complexity of determining its rank is O(n3). But in special cases we can work with graph and matroid theoretical models from which very fast and effective algorithms can be obtained. At first the case of planar square grids will be presented where they can be made rigid with diagonal rods and cables in the squares, and with long rods and cables which may be placed between any two joints of the grid. Then we will consider the one- and multi-story buildings, and finally some other results and algorithms.