On shredders and vertex connectivity augmentation

Authors:
Gilad Liberman;Zeev Nutov
Affiliations:
The Open University of Israel, Department of Computer Science, 108 Ravutski Str., POB 808, Raanana 43107, Israel;The Open University of Israel, Department of Computer Science, 108 Ravutski Str., POB 808, Raanana 43107, Israel
Venue:
Journal of Discrete Algorithms
Year:
2007

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Abstract

We consider the following problem: given a k-(node) connected graph G find a smallest set F of new edges so that the graph G+F is (k+1)-connected. The complexity status of this problem is an open question. The problem admits a 2-approximation algorithm. Another algorithm due to Jordan computes an augmenting edge set with at most @?(k-1)/2@? edges over the optimum. C@?V(G) is a k-separator (k-shredder) of G if |C|=k and the number b(C) of connected components of G-C is at least two (at least three). We will show that the problem is polynomially solvable for graphs that have a k-separator C with b(C)=k+1. This leads to a new splitting-off theorem for node connectivity. We also prove that in a k-connected graph G on n nodes the number of k-shredders with at least p components (p=3) is less than 2n/(2p-3), and that this bound is asymptotically tight.