Maxwell-independence: a new rank estimate for the 3-dimensional generic rigidity matroid

The problem of combinatorially determining the rank of the 3-dimensional bar-joint 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 independent in its d-dimensional generic rigidity matroid only if (a) the number of edges |E| ≤ d|V | − ( d+1 2 ) , and (b) this holds for every induced subgraph with at least d vertices. We call such graphs 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 Tibór Jordán at the 2008 rigidity workshop at BIRS [4]. Along the way, we construct subgraphs (1) that yield alternative formulae for a rank upper bound for Maxwellindependent 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.