  A rigidity circuit (in 2D) is a minimal dependent set in the rigidity
matroid, i.e. a minimal graph supporting a non-trivial stress in any generic
placement of its vertices in $\mathbb R^2$. Any rigidity circuit on $n\geq 5$
vertices can be obtained from rigidity circuits on a fewer number of vertices
by applying the combinatorial resultant (CR) operation. The inverse operation
is called a combinatorial resultant decomposition (CR-decomp). Any rigidity
circuit on $n\geq 5$ vertices can be successively decomposed into smaller
circuits, until the complete graphs $K_4$ are reached. This sequence of
CR-decomps has the structure of a rooted binary tree called the combinatorial
resultant tree (CR-tree).
  A CR-tree encodes an elimination strategy for computing circuit polynomials
via Sylvester resultants. Different CR-trees lead to elimination strategies
that can vary greatly in time and memory consumption. It is an open problem to
establish criteria for optimal CR-trees, or at least to characterize those
CR-trees that lead to good elimination strategies. In [12] we presented an
algorithm for enumerating CR-trees where we give the algorithms for decomposing
3-connected rigidity circuits in polynomial time. In this paper we focus on
those circuits that are not 3-connected, which we simply call 2-connected.
  In order to enumerate CR-decomps of 2-connected circuits $G$, a brute force
exp-time search has to be performed among the subgraphs induced by the subsets
of $V(G)$. This exp-time bottleneck is not present in the 3-connected case. In
this paper we will argue that we do not have to account for all possible
CR-decomps of 2-connected rigidity circuits to find a good elimination
strategy; we only have to account for those CR-decomps that are a 2-split, all
of which can be enumerated in polynomial time. We present algorithms and
computational evidence in support of this heuristic.
