TY - JOUR

T1 - On shredders and vertex connectivity augmentation

AU - Liberman, Gilad

AU - Nutov, Zeev

PY - 2007/3

Y1 - 2007/3

N2 - 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 Jordán 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 2 n / (2 p - 3), and that this bound is asymptotically tight.

AB - 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 Jordán 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 2 n / (2 p - 3), and that this bound is asymptotically tight.

KW - Exact/approximation algorithms

KW - Node-connectivity augmentation

KW - Shredders

UR - http://www.scopus.com/inward/record.url?scp=33947607980&partnerID=8YFLogxK

U2 - 10.1016/j.jda.2006.03.005

DO - 10.1016/j.jda.2006.03.005

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AN - SCOPUS:33947607980

SN - 1570-8667

VL - 5

SP - 91

EP - 101

JO - Journal of Discrete Algorithms

JF - Journal of Discrete Algorithms

IS - 1

ER -