  We introduce the idea of Assur graphs, a concept originally developed and
exclusively employed in the literature of the kinematics community. The paper
translates the terminology, questions, methods and conjectures from the
kinematics terminology for one degree of freedom linkages to the terminology of
Assur graphs as graphs with special properties in rigidity theory. Exploiting
recent works in combinatorial rigidity theory we provide mathematical
characterizations of these graphs derived from minimal linkages. With these
characterizations, we confirm a series of conjectures posed by Offer Shai, and
offer techniques and algorithms to be exploited further in future work.
