Mathematics – Combinatorics
Scientific paper
2010-10-18
Mathematics
Combinatorics
To appear in SODA'11. Added proofs omitted from the proceedings version
Scientific paper
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 Erdos-Renyi random graph G(n,c/n), then there exists a sharp threshold for a giant rigid component to emerge. For c < c_2, w.h.p. all rigid components span one, two, or three vertices, and when c > c_2, w.h.p. there is a giant rigid component. The constant c_2 \approx 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.
Kasiviswanathan Shiva Prasad
Moore Cristopher
Theran Louis
No associations
LandOfFree
The rigidity transition in random graphs does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with The rigidity transition in random graphs, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The rigidity transition in random graphs will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-306066