Minimally rigid graphs can be recognized and embedded in the plane efficiently, i.e. in polynomial time. There is also an efficient randomized parallel algorithm, i.e. in RNC. We present NC-algorithms to recognize whether one-crossing-minor-free graphs are minimally rigid. In the special case of $K_{3,3}$-free graphs, we also compute an infinitesimally rigid embedding in NC.

Minimally rigid graphs can be recognized and embedded in the plane efficiently, i.e. in polynomial time. There is also an efficient randomized parallel algorithm, i.e. in RNC. We present NC-algorithms to recognize whether one-crossing-minor-free graphs are minimally rigid. In the special case of $K_{3,3}$-free graphs, we also compute an infinitesimally rigid embedding in NC.