  Consider a graph G with n vertices. In this paper we study geometric
conditions for an n-tuple of points in R^d to admit a tensegrity with
underlying graph G. We introduce and investigate a natural stratification,
depending on G, of the configuration space of all n-tuples in R^d. In
particular we find surgeries on graphs that give relations between different
strata. Based on numerous examples we give a description of geometric
conditions defining the strata for plane tensegrities, we conjecture that the
list of such conditions is sufficient to describe any stratum. We conclude the
paper with particular examples of strata for tensegrities in the plane with a
small number of vertices.
