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The algebraic combinatorial approach for low-rank matrix completion
TLDR
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
Generic combinatorial rigidity of periodic frameworks
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 rigidityExpand
Henneberg constructions and covers of cone-Laman graphs
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 withExpand
Slider-Pinning Rigidity: a Maxwell–Laman-Type Theorem
TLDR
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
Graded Sparse Graphs and Matroids
TLDR
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
Sparsity-certifying Graph Decompositions
TLDR
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
Algebraic Matroids with Graph Symmetry
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 columnExpand
The rigidity transition in random graphs
TLDR
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
Topological designs
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 theExpand
Finding and Maintaining Rigid Components
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 aExpand
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