This work presents probability-one algorithms to decide whether a particular entry of the matrix can be completed, and describes methods to complete that entry from a few others, and to estimate the error which is incurred by any method completing that entry.Expand

Abstract We give a combinatorial characterization of generic minimal rigidity for planar periodic frameworks . The characterization is a true analogue of the Maxwell–Laman Theorem from rigidity… Expand

We give Henneberg-type constructions for three families of sparse colored graphs arising in the rigidity theory of periodic and other forced symmetric frameworks. The proof method, which works with… Expand

This work defines and study slider-pinning rigidity through direction-slider networks, which are a generalization of Whiteley’s direction networks, giving a complete combinatorial characterization.Expand

A new family called graded sparse graphs is defined, arising from generically pinned (completely immobilized) bar-and-joint frameworks and it is proved that they also form matroids.Expand

The authors' colored pebbles generalize and strengthen the previous results of Lee and Streinu and give a new proof of the Tutte-Nash-Williams characterization of arboricity, and present a new decomposition that certifies sparsity based on the (k, ℓ)-pebble game with colors.Expand

This paper studies the properties of two kinds of matroids: (a) algebraic matroids and (b) finite and infinite matroids whose ground set have some canonical symmetry, for example row and column… Expand

It is shown that if this graph is an Erdős-Rényi random graph G(n, c/n), then there exists a sharp threshold for a giant rigid component to emerge, and it is proved that it spans a (1 − o(1))-fraction of the vertices in the (3 + 2)-core.Expand

We give an exponential upper and a quadratic lower bound on the number of pairwise non-isotopic simple closed curves which can be placed on a closed surface of genus $$g$$ such that any two of the… Expand

We give the first complete analysis that the complexity of finding and maintaining rigid components of planar bar-and-joint frameworks and arbitrary d-dimensional body-and-bar frameworks, using a… Expand