Maxwell-independence: a new rank estimate for 3D rigidity matroids

TL;DR

This paper introduces a new approach to estimate the rank of 3D rigidity matroids using Maxwell-independence, providing bounds and insights that improve understanding of 3D rigidity properties.

Contribution

It establishes that maximal Maxwell-independent sets in 3D graphs have size at least the matroid's rank, extending 2D results and offering new formulas and bounds.

Findings

Maximal Maxwell-independent sets have size ≥ the 3D rigidity matroid rank.

Constructs subgraphs for alternative rank bounds and independent sets.

Provides simpler proofs for existing algorithms' correctness.

Abstract

The problem of combinatorially determining the rank of the 3-dimensional bar-joint {\em rigidity matroid} of a graph is an important open problem in combinatorial rigidity theory. Maxwell's condition states that the edges of a graph $G=(V, E)$ are {\em independent} in its $d$-dimensional generic rigidity matroid only if $(a)$ the number of edges $|E|$ $\le$ $d|V| - {d+1\choose 2}$, and $(b)$ this holds for every induced subgraph with at least $d$ vertices. We call such graphs {\em Maxwell-independent} in $d$ dimensions. Laman's theorem shows that the converse holds for $d=2$ and thus every maximal Maxwell-independent set of $G$ has size equal to the rank of the 2-dimensional generic rigidity matroid. While this is false for $d=3$, we show that every maximal, Maxwell-independent set of a graph $G$ has size at least the rank of the 3-dimensional generic rigidity matroid of $G$. This answers a question posed by Tib\'or Jord\'an at the 2008 rigidity workshop at BIRS \cite{bib:birs}. Along the way, we construct subgraphs (1) that yield alternative formulae for a rank upper bound for Maxwell-independent graphs and (2) that contain a maximal (true) independent set. We extend this bound to special classes of non-Maxwell-independent graphs. One further consequence is a simpler proof of correctness for existing algorithms that give rank bounds.