Cayley Configuration Spaces of 1-dof Tree-decomposable Linkages, Part   II: Combinatorial Characterization of Complexity
  We continue to study Cayley configuration spaces of 1-dof linkages in 2D begun in Part I of this paper, i.e. the set of attainable lengths for a non-edge. In Part II, we focus on the algebraic complexity of describing endpoints of the intervals in the set, i.e., the Cayley complexity.   Specifically, We focus on Cayley configuration spaces of a natural class of 1-dof linkages, called 1-dof tree-decomposable linkages. The underlying graphs G satisfy the following: for some base non-edge f, G \cup f is quadratic-radically solvable (QRS), meaning that G \cup f is minimally rigid, and given lengths \bar{l} of all edges, the corresponding linkage (G \cup f, \bar{l}) can be simply realized by ruler and compass starting from f. It is clear that the Cayley complexity only depends on the graph G and possibly the non-edge f. Here we ask whether the Cayley complexity depends on the choice of a base non-edge f. We answer this question in the negative, thereby showing that low Cayley complexity is a property of the graph G (independent of the non-edge f).   Then, we give a simple characterization of graphs with low Cayley complexity, leading to an efficient algorithmic characterization, i.e. an efficient algorithm for recognizing such graphs.   Next, we show a surprising result that (graph) planarity is equivalent to low Cayley complexity for a natural subclass of 1-dof triangle-decomposable graphs. While this is a finite forbidden minor graph characterization of low Cayley complexity, we provide counterexamples showing impossibility of such finite forbidden minor characterizations when the above subclass is enlarged. 
Meera Sitharam, Menghan Wang, Heping Gao
cs.CG