Characterizing graphs with convex and connected cayley configuration spaces

Abstract

We define and study exact, efficient representations of realization spaces Euclidean Distance Constraint Systems (EDCS), which include Linkages and Frameworks. 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-edge (squared) distances or Cayley parameters. The set of realizable distance assignments to the chosen parameters yields a parametrized Cayley configuration space. Our notion of efficiency is based on the convexity and connectedness of the Cayley configuration space, as well as algebraic complexity of sampling realizations, i.e., sampling the Cayley configuration space and obtaining a realization from the sample (parametrized) configuration. Significantly, we give purely graph-theoretic, forbidden minor characterizations for 2D and 3D EDCS that capture (i) the class of graphs that always admit efficient Cayley configuration spaces and (ii) the possible choices of representation parameters that yield efficient Cayley configuration spaces for a given graph. We show that the easy direction of the 3D characterization extends to arbitrary dimension d and is related to the concept of d-realizability of graphs. Our results automatically yield efficient algorithms for obtaining exact descriptions of the Cayley 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 Cayley 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 Cayley 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 results related to positive semi-definiteness of Euclidean distance matrices, with (b) recent forbidden minor characterizations and algorithms related to d-realizability of graphs. 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. © 2009 Springer Science+Business Media, LLC.