Characterizing graphs with convex and connected configuration spaces

We define and study exact, efficient representations of realization spaces Euclidean Distance Constraint Systems (EDCS). These are graphs with distance assignments on the edges (frameworks) or graphs with distance interval assignments on the edges. Each representation corresponds to a choice of non-edges or Cayley parameters. The set of realizable distance assignments to the chosen parameters yields a parametrized configuration space. Our notion of efficiency is based on the convexity and connectedness of the configuration space, as well as algebraic complexity of sampling realizations, i.e., sampling the configuration space and obtaining a realization from the sample (parametrized) configuration. 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. Our results automatically yield efficient algorithms for obtaining exact descriptions of the configuration spaces and for sampling realizations, without missing extreme or boundary realizations. 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. In particular, the results presented here are the first to completely characterize EDCS that have connected, convex and efficient configuration spaces, based on precise and formal measures of efficiency. It should be noted that our results do not rely on genericity of the EDCS. Some of our proofs employ an unusual interplay of (a) classical analytic and algebraic results related to positive semi-definiteness of Euclidean distance matrices, and Cayley-Menger conditions, with (b) recent forbidden minor characterizations and algorithms related to 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.