# A response to Volgenant's addendum on the most vital edges

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## Vital Edges

Chun-Nan Hung and Lih-Hsing Hsu*

Department of Computer and Information Science, National Chiao Tung University, Hsinchu, Taiwan, Republic of China

Ting-Yi Sung

Institute of Information Science, Academia Sinica, Taipei, Taiwan, Republic of China

Let G = ( U U V, E ) be a weighted bipartite graph having an edge weight u’, 2 0 for each e in E . An edge is

called a must vitul edge if its removal from G results in

the largest decrease in the total weight of the maximum weighted matching. In [ I ] , an O( n 3 ) algorithm was pre- sented to obtain the most vital edges. In [ 3 ] , Volgenant pointed out that the most vital edges can also be found using the driul solutions of the linear assignment problem [ 21. We were unaware of this result and studied this prob- lem from different point of view. In our paper, we first gave characterization of the most vital edges in Lemma 1 and the effect of deleting a matched edge which is a candidate for the most vital edges in Lemma 6. Our mo- tivation was to study the effect on the cost of any com- binatonal optimization problem subject to the deletion of an edge in turn.

* To whom correspondence should be addressed

NETWORKS, Vol. 27 (1 996) 255

(c 1996 John Wiley & Sons, Inc.

We did not necessarily know the shortest distances ui

from an arbitrarily chosen vertex to all other vertices i,

i.e., the dual solution specified in [ 2 ] . Thus, we simply chose Floyd’s algorithm rather than Dijkstra’s algorithm to accommodate negative edge weights for solving short- est-path problems.

REFERENCES

[ 1 1 C. N. Hung, L. H. Hsu, and T. Y. Sung, The most vital edges of matchings in a bipartite graph, Networks 23 ( 1993) 309-3 13.

G. Kindervater, A. Volgenant, G. de Leve, and V. van Gijlswijk, On dual solutions of the linear assignment problem. Eur. J. of Oper. Rex 19 (1985) 76-81.

A. Volgenant, An addendum to the most vital edges of matching in a bipartite graph. Networks, to appear. [ 2 ]

[ 3 ]

CCC 0028-3045/96/030255-01 255

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