# On Rooted Node-Connectivity Problems

DOI: 10.1007/s00453-001-0017-7

- Cite this article as:
- Cheriyan, J., Jordán, T. & Nutov, Z. Algorithmica (2001) 30: 353. doi:10.1007/s00453-001-0017-7

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## Abstract.

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* -node-connectivity.

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 minimum-weight 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+2q-1)/k}* . Here *k* is the maximum node requirement, and *q* is the number of positive node requirements.