Sudden emergence of a giant k-core in a random graph
Journal of Combinatorial Theory Series B
An algorithm for two-dimensional rigidity percolation: the pebble game
Journal of Computational Physics
The Size of the Giant Component of a Random Graph with a Given Degree Sequence
Combinatorics, Probability and Computing
The pure literal rule threshold and cores in random hypergraphs
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Discrete & Computational Geometry
The cores of random hypergraphs with a given degree sequence
Random Structures & Algorithms
A simple solution to the k-core problem
Random Structures & Algorithms - Proceedings from the 12th International Conference “Random Structures and Algorithms”, August1-5, 2005, Poznan, Poland
The k-orientability thresholds for Gn, p
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
The 2-dimensional rigidity of certain families of graphs
Journal of Graph Theory
Slider-Pinning Rigidity: a Maxwell–Laman-Type Theorem
Discrete & Computational Geometry
A critical point for random graphs with a given degree sequence
Random Structures & Algorithms
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As we add rigid bars between points in the plane, at what point is there a giant (linear-sized) rigid component, which can be rotated and translated, but which has no internal flexibility? If the points are generic, this depends only on the combinatorics of the graph formed by the bars. We show 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. For c c2, w.h.p. all rigid components span one, two, or three vertices, and when c c2, w.h.p. there is a giant rigid component. The constant c2 ≈ 3.588 is the threshold for 2-orientability, discovered independently by Fernholz and Ramachandran and Cain, Sanders, and Wormald in SODA'07. We also give quantitative bounds on the size of the giant rigid component when it emerges, proving that it spans a (1 − o(1))-fraction of the vertices in the (3 + 2)-core. Informally, the (3 + 2)-core is maximal induced subgraph obtained by starting from the 3-core and then inductively adding vertices with 2 neighbors in the graph obtained so far.