Cayley Configuration Spaces of 1-dof Tree-decomposable Linkages, Part II: Combinatorial Characterization of Complexity

We continue to study Cayley configuration spaces of 1-degree-of-freedom (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 (defined in Part I, Section 2). The underlying graphs G satisfy the following: for some base non-edge f , G ∪ f is quadratically-radically solvable (QRS), meaning that G∪f is minimally rigid, and given lengths l̄ of all edges, the corresponding linkage (G∪f, l̄) 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 alone (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 tree-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. Email addresses: [email protected] (Meera Sitharam), [email protected] (Menghan Wang), [email protected] (Heping Gao). 1 Present address: SDE II at Microsoft, Bellevue, Washington, USA Preprint submitted to Journal of Symbolic Computation November 8, 2012