  We define and study exact, efficient representations of realization spaces
Euclidean Distance Constraint Systems (EDCS), which includes Linkages and
Frameworks. Each representation corresponds to a choice of Cayley parameters
and yields a different parametrized configuration space. Significantly, we give
purely graph-theoretic, forbidden minor characterizations that capture (i) the
class of graphs that always admit efficient configuration spaces and (ii) the
possible choices of representation parameters that yield efficient
configuration spaces for a given graph. In addition, our results are tight: we
show counterexamples to obvious extensions. This is the first step in a
systematic and graded program of combinatorial characterizations of efficient
configuration spaces. We discuss several future theoretical and applied
research directions. Some of our proofs employ an unusual interplay of (a)
classical analytic results related to positive semi-definiteness of Euclidean
distance matrices, with (b) recent forbidden minor characterizations and
algorithms related to the notion of d-realizability of EDCS. We further
introduce a novel type of restricted edge contraction or reduction to a graph
minor, a "trick" that we anticipate will be useful in other situations.
