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.