Computing Vertex Connectivity: New Bounds from Old Techniques

The vertex connectivity κ of a graph is the smallest number of vertices whose deletion separates the graph or makes it trivial. We present the fastest known deterministic algorithm for finding the vertex connectivity and a corresponding separator. The time for a digraph having n vertices and m edges...

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Bibliographic Details
Published in:Journal of algorithms Vol. 34; no. 2; pp. 222 - 250
Main Authors: Henzinger, Monika R., Rao, Satish, Gabow, Harold N.
Format: Journal Article
Language:English
Published: San Diego, CA Elsevier Inc 01.02.2000
Elsevier
Subjects:
Combinatorics
Combinatorics. Ordered structures
Exact sciences and technology
Graph theory
Mathematics
Sciences and techniques of general use
Distance function
Vertex connectivity
Connectedness
Graph theory
Algorithm
Directed graph
Weighted graph
Non directed graph
ISSN:0196-6774, 1090-2678
Online Access:Get full text
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Summary:The vertex connectivity κ of a graph is the smallest number of vertices whose deletion separates the graph or makes it trivial. We present the fastest known deterministic algorithm for finding the vertex connectivity and a corresponding separator. The time for a digraph having n vertices and m edges is O(min{κ3+n,κn}m); for an undirected graph the term m can be replaced by κn. A randomized algorithm finds κ with error probability 1/2 in time O(nm). If the vertices have nonnegative weights the weighted vertex connectivity is found in time O(κ1nmlog(n2/m)) where κ1≤m/n is the unweighted vertex connectivity or in expected time O(nmlog(n2/m)) with error probability 1/2. The main algorithm combines two previous vertex connectivity algorithms and a generalization of the preflow-push algorithm of Hao and Orlin (1994, J. Algorithms17, 424–446) that computes edge connectivity.
ISSN:0196-6774
1090-2678
DOI:10.1006/jagm.1999.1055