Rigid Two-Dimensional Frameworks with Two Coincident Points

Let G = ( V , E ) be a graph and u, v ∈ V be two distinct vertices. We give a necessary and sufficient condition for the existence of an infinitesimally rigid two-dimensional bar-and-joint framework ( G , p ), in which the positions of u and v coincide. We also determine the rank function of the corresponding modified generic rigidity matroid on ground-set E . The results lead to efficient algorithms for testing whether a graph has such a coincident realization with respect to a designated vertex pair and, more generally, for computing the rank of G in the matroid.