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Tight Bounds on Vertex Connectivity Under Sampling

Authors:

Keren Censor-Hillel,

Mohsen Ghaffari,

George Giakkoupis,

Bernhard Haeupler, and

Fabian KuhnAuthors Info & Claims

ACM Transactions on Algorithms (TALG), Volume 13, Issue 2

Article No.: 19, Pages 1 - 26

https://doi.org/10.1145/3086465

Published: 29 May 2017 Publication History

Abstract

A fundamental result by Karger [10] states that for any λ-edge-connected graph with n nodes, independently sampling each edge with probability p = Ω(log (n)/λ) results in a graph that has edge connectivity Ω(λp), with high probability. This article proves the analogous result for vertex connectivity, when either vertices or edges are sampled. We show that for any k-vertex-connected graph G with n nodes, if each node is independently sampled with probability p=Ω(√log(n)/k), then the subgraph induced by the sampled nodes has vertex connectivity Ω(kp²), with high probability. If edges are sampled with probability p = Ω(log (n)/k), then the sampled subgraph has vertex connectivity Ω(kp), with high probability. Both bounds are existentially optimal.

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Cited By

Chandra SChang YDory MGhaffari MLeitersdorf DAgrawal KPetrank E(2024)Fast Broadcast in Highly Connected NetworksProceedings of the 36th ACM Symposium on Parallelism in Algorithms and Architectures10.1145/3626183.3659959(331-343)Online publication date: 17-Jun-2024
https://dl.acm.org/doi/10.1145/3626183.3659959
Aigner‐Horev EHefetz DKrivelevich M(2023)Cycle lengths in randomly perturbed graphsRandom Structures & Algorithms10.1002/rsa.21170Online publication date: 20-Jun-2023
https://doi.org/10.1002/rsa.21170

Index Terms

Tight Bounds on Vertex Connectivity Under Sampling
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
  2. Probability and statistics

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Published In

cover image ACM Transactions on Algorithms

ACM Transactions on Algorithms Volume 13, Issue 2

Special Issue on SODA'15 and Regular Papers

April 2017

316 pages

ISSN:1549-6325

EISSN:1549-6333

DOI:10.1145/3040971

Editor:
Aravind Srinivasan
University of Maryland, USA

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Copyright © 2017 ACM.

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 29 May 2017

Accepted: 01 April 2017

Revised: 01 March 2017

Received: 01 June 2015

Published in TALG Volume 13, Issue 2

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Cited By

Chandra SChang YDory MGhaffari MLeitersdorf DAgrawal KPetrank E(2024)Fast Broadcast in Highly Connected NetworksProceedings of the 36th ACM Symposium on Parallelism in Algorithms and Architectures10.1145/3626183.3659959(331-343)Online publication date: 17-Jun-2024
https://dl.acm.org/doi/10.1145/3626183.3659959
Aigner‐Horev EHefetz DKrivelevich M(2023)Cycle lengths in randomly perturbed graphsRandom Structures & Algorithms10.1002/rsa.21170Online publication date: 20-Jun-2023
https://doi.org/10.1002/rsa.21170

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