On Rooted NodeConnectivity Problems
 J. Cheriyan,
 T. Jordán,
 Z. Nutov
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Let G be a graph which is k outconnected from a specified root node r , that is, G has k openly disjoint paths between r and v for every node v . We give necessary and sufficient conditions for the existence of a pair rv,rw of edges for which replacing these edges by a new edge vw gives a graph that is k outconnected from r . This generalizes a theorem of Bienstock et al. on splitting off edges while preserving k nodeconnectivity.
We also prove that if C is a cycle in G such that each edge in C is critical with respect to k outconnectivity from r , then C has a node v , distinct from r , which has degree k . This result is the rooted counterpart of a theorem due to Mader.
We apply the above results to design approximation algorithms for the following problem: given a graph with nonnegative edge weights and node requirements c _{ u } for each node u , find a minimumweight subgraph that contains max {c _{ u } ,c _{ v } } openly disjoint paths between every pair of nodes u,v . For metric weights, our approximation guarantee is 3 . For uniform weights, our approximation guarantee is \min{ 2, (k+2q1)/k} . Here k is the maximum node requirement, and q is the number of positive node requirements.
 Title
 On Rooted NodeConnectivity Problems
 Journal

Algorithmica
Volume 30, Issue 3 , pp 353375
 Cover Date
 20010101
 DOI
 10.1007/s0045300100177
 Print ISSN
 01784617
 Online ISSN
 14320541
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Key words. Graph connectivity, k Connectivity, k Outconnectivity, Splittingoff theorems, NPhard problems, Approximation algorithms, Metric costs, Uniform costs.
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 Authors

 J. Cheriyan ^{(A1)}
 T. Jordán ^{(A2)}
 Z. Nutov ^{(A3)}
 Author Affiliations

 A1. Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1. jcheriyan@math.uwaterloo.ca., CA
 A2. Department of Operations Research, Eőtvős University, 1053 Budapest, Hungary. jordan@cs.elte.hu., HU
 A3. Open University, Tel Aviv, Israel. nutov@oumail.openu.ac.il., IL