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We continue to study Cayley configuration spaces of 1-dof linkages in 2Dbegun in Part I of this paper, i.e. the set of attainable lengths for anon-edge. In Part II, we focus on the algebraic complexity of describingendpoints of the intervals in the set, i.e., the Cayley complexity.  Specifically, We focus on Cayley configuration spaces of a natural class of1-dof linkages, called 1-dof tree-decomposable linkages. The underlying graphsG satisfy the following: for some base non-edge f, G \cup f isquadratic-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 isclear that the Cayley complexity only depends on the graph G and possibly thenon-edge f. Here we ask whether the Cayley complexity depends on the choice ofa base non-edge f. We answer this question in the negative, thereby showingthat low Cayley complexity is a property of the graph G (independent of thenon-edge f).  Then, we give a simple characterization of graphs with low Cayley complexity,leading to an efficient algorithmic characterization, i.e. an efficientalgorithm for recognizing such graphs.  Next, we show a surprising result that (graph) planarity is equivalent to lowCayley complexity for a natural subclass of 1-dof triangle-decomposable graphs.While this is a finite forbidden minor graph characterization of low Cayleycomplexity, we provide counterexamples showing impossibility of such finiteforbidden minor characterizations when the above subclass is enlarged.

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