Bounded Direction-Length Frameworks

A direction-length framework is a pair (G, p), where G = (V ;D,L) is a ‘mixed’ graph whose edges are labeled as ‘direction’ or ‘length’ edges, and p is a map from V to R for some d. The label of an edge uv represents a direction or length constraint between p(u) and p(v). Let G be obtained from G by adding, for each length edge e of G, a direction edge with the same end vertices as e. We show that (G, p) is bounded if and only if (G, p) is infinitesimally rigid. This gives a characterisation of when (G, p) is bounded in terms of the rank of the rigidity matrix of (G, p). We use this to characterise when a mixed graph is generically bounded in R. As an application we deduce that if (G, p) is a globally rigid generic framework with at least two length edges and e is a length edge of G then (G− e, p) is bounded.