ABSTRACT
This chapter considers the properties of generic frameworks which are weaker than global rigidity. It characterizes the global rigidity in all dimensions for mixed graphs in which every pair of adjacent vertices is connected by both a length and a direction edge. Inductive constructions for graphs which use global rigidity preserving operations are frequently used, both to prove that certain families of graphs are globally rigid and to analyze the global rigidity of a particular graph. Vertex transitive graphs which are rigid or globally rigid were characterized by B. Jackson, B. Servatius, and H. Servatius. One can consider higher degrees of redundant rigidity with respect to edge removal as well as similar notions for global rigidity. An infinitesimally rigid framework is regular valued if all equivalent frameworks are infinitesimally rigid. It is known that an infinitesimally rigid, regular valued framework has only finitely many equivalent and pairwise non-congruent realizations.
