Lower bounds on the number of realizations of rigid graphs

Computing the number of realizations of a minimally rigid graph is a notoriously difficult problem. Towards this goal, for graphs that are minimally rigid in the plane, we take advantage of a recently published algorithm, which is the fastest available method, although its complexity is still exponential. Combining computational results with the theory of constructing new rigid graphs by gluing, we give a new lower bound on the maximal possible number of (complex) realizations for graphs with a given number of vertices. We extend these ideas to rigid graphs in three dimensions and we derive similar lower bounds, by exploiting data from extensive Gr"obner basis computations.