**A Response to Volgenant’s Addendum **

**on the Most **

**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 ]

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