ISAAC 1994: Algorithms and Computation pp 217-225

# Minimum augmentation to k-edge-connect specified vertices of a graph

• Satoshi Taoka
• Toshimasa Watanabe
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 834)

## Abstract

The k-edge-connectivity augmentation problem for a specified set of vertices (kECA-SV for short) is defined by “Given a graph G=(V, E) and a subset Γ$$\subseteq$$V, find a minimum set E′ of edges,each connecting distinct vertices of V, such that G′=(V, EE′) has at least k edge-disjoint paths between any pair of vertices in Γ”. We propose an O(λ2¦V¦(¦V¦+¦Γ¦log λ)+¦E¦) algorithm for (λ + 1)ECA-SV with Γ(V), where λ is the edge-connectivity of Γ (the cardinality of a minimum cut separating two vertices of Γ). Also mentioned is an OV¦ log ¦V¦+¦E¦) algorithm for a special case where λ is equal to the edge-connectivity of G.

## Preview

Unable to display preview. Download preview PDF.

## References

1. 1.
Eswaran, K. P. and Tarjan, R. E.: Augmentation problems, SIAM J. Comput, 5 (1976), 653–655.
2. 2.
Even, S.: Graph Algorithms, Pitman, London (1979).Google Scholar
3. 3.
Frank, A.: Augmenting graphs to meet edge connectivity requirements, SIAM J. Discrete Mathematics, 5, 1 (1992), 25–53.
4. 4.
Gabow, H. N.: Applications of a Poset Representation to Edge Connectivity and Graph Rigidity, Proc. 32nd IEEE Symposium on Foundations of Computer Science (1991), 812–821.Google Scholar
5. 5.
Karzanov, A. V. and Timofeev, E. A.: Efficient algorithm for finding all minimal edge cuts of a nonoriented graph, Cybernetics (March–April 1986), 156–162, Translated from Kibernetika, 2 (1986), pp.8–12.Google Scholar
6. 6.
Nagamochi, H. and Ibaraki, T.: Computing edge-connectivity in multigraphs and capacitated graphs, SIAM J. Discrete Mathematics, 5 (1992), 54–66.
7. 7.
Nagamochi, H. and Ibaraki, T.: A linear time algorithm for finding a sparse k-connected spanning subgraph of a k-connected graph, Algorithmica, 7 (1992), 583–596.
8. 8.
Nagamochi, H. and Watanabe, T.: Computing k-Edge-Connected Components of a Multigraphs, Trans. IEICE of Japan, E76-A, 4 (1993), 513–517.Google Scholar
9. 9.
Naor, D., Gusfield, D. and Martel, C.: A Fast Algorithm for Optimally Increasing the Edge Connectivity, Proc. 31st Annual IEEE Symposium on Foundations of Computer Science (1990), 698–707.Google Scholar
10. 10.
Takafuji, D., Taoka, S. and Watanabe, T.: Simplicity-preserving augmentation to 4-edge-connect a graph, IPSJ SIG Notes, AL-33-5 (May 1993), 33–40.Google Scholar
11. 11.
Taoka, S., Takafuji, D. and Watanabe, T.: Simplicity-preserving augmentation of the edge-connectivity of a graph, Tech. Rep. of IEICE of Japan, COMP93-73 (January 1994), 49–56.Google Scholar
12. 12.
Taoka, S. and Watanabe, T.: Smallest augmentation to k-edge-connect all specified vertices in a graph, IPSJ SIG Notes, AL-38-3 (March 1994), 17–24.Google Scholar
13. 13.
Watanabe, T.: An efficient way for edge-connectivity augmentation, Tec. Rep. ACT-76-UILU-ENG-87-2221, Coordinated Science Lab., University of Illinois at Urbana, Urbana, IL 61801 (April 1987). Also presented at Eighteenth Southeastern International Conference on Combinatorics, Graph Theory, Computing, No.15, Boca Raton, FL, U.S.A., February 1987.Google Scholar
14. 14.
Watanabe, T., Higashi, Y. and Nakamura, A.: An approach to robust networks construction from graph augmentation problems, Proc. of 1990 IEEE International Symposium on Circuits and Systems (May 1990), 2861–2864.Google Scholar
15. 15.
Watanabe, T. and Nakamura, A.: Edge-connectivity augmentation problems, J. Comput. System Sci., 35, 1 (1987), 96–144.Google Scholar
16. 16.
Watanabe, T., Taoka, S. and Mashima, T.: Minimum-cost augmentation to 3-edge-connect all specified vertices in a graph, Proc.1993 IEEE International Symposium on Circuits and Systems (May 1993), 2311–2314.Google Scholar
17. 17.
Watanbe, T. and Yamakado, M.: A linear time algorithm for smallest augmentation to 3-edge-connect a graph, Trans. IEICE of Japan, E76-A, 4 (1993), 518–531.Google Scholar