Combinatorial Algorithm for a Lower Bound on Frame Rigidity

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A classic problem is that of computing the rigidity of a network of rigid bars or, more formally, computing the degrees of freedom of a frame, a graph with a generic straight-line embedding in Euclidean space. More recently, in trying to explain the effect of the internal structure of glassy materials on their rigidity, researchers have investigated the relationship between the structure of network models and the number of degrees of freedom. It would thus be valuable to have an efficient, purely combinatorial algorithm to compute the degrees of freedom of an arbitrary frame. For frames in two dimensions, there are several such algorithms, all based on a theorem of Laman. However, despite considerable effort, no one has yet found such an algorithm for frames in three or more dimensions. Here, a new combinatorial approach, similar to ear decomposition, is introduced and shown to give a practical algorithm for computing a nontrivial lower bound on the number of degrees of freedom in three dimensions. Results of computational studies on an important class of network models of glasses are given, suggesting that the resulting bound is close to the exact answer in the cases of interest. The algorithm has been implemented in $O( n )$ space and $O( n^2 )$ time for bounded-degree networks with n vertices and has already provided new results of interest on these networks.