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

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.