Ear‐decompositions, minimally connected matroids and rigid graphs

Abstract

AbstractWe prove that if the two‐dimensional rigidity matroid of a graph on at least seven vertices is connected, and is minimal with respect to this property, then has at most edges. This bound, which is best possible, extends Dirac's bound on the size of minimally 2‐connected graphs to dimension two. The bound also sharpens the general upper bound of Murty for the size of minimally connected matroids in the case when the matroid is a rigidity matroid of a graph. Our proofs rely on ear‐decompositions of connected matroids and on a new lower bound on the size of the largest circuit in a connected rigidity matroid, which may be of independent interest. We use these results to determine the tight upper bound on the number of edges in a minimally redundantly rigid graph in two dimensions. Furthermore, as an application of our proof methods, we give a new proof for Murty's theorem.