Enumerating combinatorial resultant trees

Abstract

A 2D rigidity circuit is a minimal graph $G=(V,E)$ supporting a non-trivial stress in any generic placement of its vertices in the Euclidean plane. All 2D rigidity circuits can be constructed from $K_4$ graphs using combinatorial resultant (CR) operations. A combinatorial resultant tree (CR-tree) is a rooted binary tree capturing the structure of such a construction.

The CR operation has a specific algebraic interpretation, where an essentially unique circuit polynomial is associated to each circuit graph. Performing Sylvester resultant operations on these polynomials is in one-to-one correspondence with CR operations on circuit graphs. This mixed combinatorial/algebraic approach led recently to an effective algorithm for computing circuit polynomials. Its complexity analysis remains an open problem, but it is known to be influenced by the depth and shape of CR-trees in ways that have only partially been investigated.

In this paper, we present an effective algorithm for enumerating all the CR-trees of a given circuit with n vertices. Our algorithm has been fully implemented in Mathematica and allows for computational experimentation with various optimality criteria in the resulting, potentially exponentially large collections of CR-trees.