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.

翻译：刚性电路（二维中）是刚性拟阵中的极小相关集，即在任意顶点一般位置放置于$\mathbb R^2$中时，能支撑非平凡应力的极小图。任何$n\geq 5$个顶点的刚性电路均可通过应用组合合成（CR）操作，从更少顶点数的刚性电路获得。逆操作称为组合合成分解（CR-decomp）。任何$n\geq 5$个顶点的刚性电路可被逐步分解为更小的电路，直至达到完全图$K_4$。这一CR-decomp序列具有称为组合合成树（CR-tree）的有根二叉树结构。CR-tree编码了通过Sylvester结式计算电路多项式的消去策略。不同的CR-tree导致的消去策略在时间和内存消耗上可能存在巨大差异。如何建立最优CR-tree的判据，或至少刻画那些能产生良好消去策略的CR-tree，是一个开放问题。在文献[12]中，我们提出了一种枚举CR-tree的算法，其中给出了多项式时间内分解3-连通刚性电路的算法。本文聚焦于非3-连通的电路，简称为2-连通。为枚举2-连通电路$G$的CR-decomp，需在$V(G)$的子集所诱导的子图中进行指数时间暴力搜索。这一指数时间瓶颈在3-连通情形中并不存在。本文将论证：为找到良好消去策略，我们无需考虑2-连通刚性电路的所有可能CR-decomp，只需考虑那些为2-分裂（2-split）的CR-decomp，而所有这些分解均可在多项式时间内枚举。我们给出了支持这一启发式策略的算法与计算证据。