Globally rigid circuits of the direction–length rigidity matroid

A two-dimensional mixed framework is a pair ( G , p ) , where G = ( V ; D , L ) is a graph whose edges are labeled as ‘direction’ or ‘length’ edges, and p is a map from V to R 2 . The label of an edge uv represents a direction or length constraint between p ( u ) and p ( v ) . The framework ( G , p ) is globally rigid if every framework ( G , q ) in which the direction or length between the end vertices of corresponding edges is the same as in ( G , p ) , can be obtained from ( G , p ) by a translation and, possibly, a dilation by −1. racterize the globally rigid generic mixed frameworks ( G , p ) for which the edge set of G is a circuit in the associated direction–length rigidity matroid. We show that such a framework is globally rigid if and only if each 2-separation S of G is ‘direction balanced’, i.e. each ‘side’ of S contains a direction edge. Our result is based on a new inductive construction for the family of edge-labeled graphs which satisfy these hypotheses. We also settle a related open problem posed by Servatius and Whiteley concerning the inductive construction of circuits in the direction–length rigidity matroid.