Better Upper Bounds on the Rank of the 3D Rigidity Matroid of a General Graph

It is a long open problem to combinatorially characterize the 3D bar-joint rigidity of graphs. The problem is at the intersection of combinatorics and algebraic geometry, and crops up in practical algorithmic applications ranging from mechanical computer aided design to molecular modeling. The problem is equivalent to combinatorially determining the generic rank of the 3D bar-joint rigidity matrix of a graph G. The k-dimensional bar-joint rigidity matrix of a graph G = (V,E), denoted R(G), is a matrix of indeterminates p1(v), p2(v), . . . pk(v). These represent the coordinate position p(v) ∈ R of the joint corresponding to a vertex v ∈ V . The matrix has one row for each edge e ∈ E and k columns for each vertex v ∈ V . The row corresponding to e = (u, v) ∈ E represents the bar from p(u) to p(v) and has k non-zero entries p(u)− p(v) (resp. p(v)− p(u)), in the k columns corresponding to u (resp. v). A subset of edges of a graph G is said to be independent (we drop “bar-joint” from now on) in k-dimensions, when the corresponding set of rows of R(G) are generically independent. This yields the 3D rigidity matroid associated with a graph G. The graph is rigid if the number of generically independent rows or the rank of R(G) is maximal, i.e., k|V | − ( k+1 2 )