Nucleation-free independent graphs with implied nonedges

TL;DR

This paper presents inductive methods to construct independent graphs with implied nonedges that lack rigid subgraphs, aiding understanding of rigidity matroids and advancing algorithms for independence testing.

Contribution

It introduces new inductive constructions of nucleation-free independent graphs with implied nonedges, relevant to rigidity matroids and algorithmic independence decisions.

Findings

Constructed classes of flexible circuits without rigid subgraphs.

Provided insights into obstacles for the maximality conjecture.

Highlighted challenges for polynomial-time independence algorithms.

Abstract

We give inductive constructions of independent graphs that contain implied nonedges but do not contain any non-trivial rigid subgraphs, or \emph{nucleations}: some of the constructions and proofs apply to 3-dimensional abstract rigidity matroids with their respective definitions of nucleations and implied nonedges. The first motivation for the inductive constructions of this paper, which generate an especially intractable class of flexible circuits, is to illuminate further obstacles to settling Graver's maximality conjecture that the 3-dimensional generic rigidity matroid is isomorphic to Whiteley's cofactor matroid (the unique maximal matroid in which all graphs isomorphic to $K_5$ are circuits). While none of the explicit examples we provide refutes the maximality conjecture (since their properties hold in both matroids) the construction schemes are useful regardless whether the conjecture is true or false, e.g. for constructing larger (counter)examples from smaller ones. The second motivation is to make progress towards a polynomial-time algorithm for deciding independence in the abovementioned maximal matroid. Nucleation-free graphs with implied nonedges, such as the families constructed in this paper, are the key obstacles that must be dealt with for improving the current state of the art.