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Graph Connectivity

1994; Khuller, Vishkin

  • Reference work entry
Encyclopedia of Algorithms

Keywords and Synonyms

Highly connected subgraphs; Sparse certificates

Problem Definition

An undirected graph is said to be k-connected (specifically, k-vertex‐connected) if the removal of any set of \( { k-1 } \) or fewer vertices (with their incident edges) does not disconnect G. Analogously, it is k-edge‐connected if the removal of any set of \( { k-1 } \) edges does not disconnect G. Menger's theorem states that a k-vertex‐connected graph has at least k openly vertex‐disjoint paths connecting every pair of vertices. For k-edge‐connected graphs there are k edge-disjoint paths connecting every pair of vertices. The connectivity of a graph is the largest value of k for which it is k-connected. Finding the connectivity of a graph, and finding k disjoint paths between a given pair of vertices can be found using algorithms for maximum flow. An edge is said to be critical in a k-connected graph if upon its removal the graph is no longer k-connected.

The problem of finding...

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Recommended Reading

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Author information

Authors and Affiliations

  1. Computer Science Department, University of Maryland, College Park, MD, USA

    Samir Khuller

  2. Computer Science Department, University of Texas at Dallas, Richardson, TX, USA

    Balaji Raghavachari

Authors
  1. Samir Khuller
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  2. Balaji Raghavachari
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Editor information

Editors and Affiliations

  1. Department of Electrical Engineering and Computer ScienceMcCormick School of Engineering and Applied Science, Northwestern University, Evanston, IL, 60208, USA

    Ming-Yang Kao Professor of Computer Science

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© 2008 Springer-Verlag

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Khuller, S., Raghavachari, B. (2008). Graph Connectivity. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-30162-4_171

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