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Edgeconnectivity augmentation with partition constraints
 SIAM J. Discrete Mathematics
, 1999
"... When k is even the minmax formula for the partitionconstrained problem is a natural generalization of [3]. However this generalization fails when k is odd. We show that at most one more edge is needed when k is odd and we characterize the graphs that require such an extra edge. ..."
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Cited by 18 (10 self)
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When k is even the minmax formula for the partitionconstrained problem is a natural generalization of [3]. However this generalization fails when k is odd. We show that at most one more edge is needed when k is odd and we characterize the graphs that require such an extra edge.
EdgeConnectivity Augmentation Preserving Simplicity
, 1997
"... Given a simple graph G = (V; E), the goal is to find a smallest set F of new edges such that G = (V; E [ F ) is kedgeconnected and simple. Very recently this problem was shown to be NPcomplete. In this paper we prove that if OPT k P is high enough  depending on k only  then OPT k S = OPT ..."
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Cited by 15 (8 self)
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Given a simple graph G = (V; E), the goal is to find a smallest set F of new edges such that G = (V; E [ F ) is kedgeconnected and simple. Very recently this problem was shown to be NPcomplete. In this paper we prove that if OPT k P is high enough  depending on k only  then OPT k S = OPT k P holds, where OPT k S (OPT k P ) is the size of an optimal solution of the augmentation problem with (without) the simplicitypreserving requirement, respectively. Furthermore, OPT k S \Gamma OPT k P g(k) holds for a certain (quadratic) function of k. Based on these facts an algorithm is given which computes an optimal solution in time O(n 4 ) for any fixed k. Some of these results are extended to the case of nonuniform demands, as well. 1 Introduction In the last decade several graph augmentation problems have been investigated. Especially the connectivity augmentation problems attracted considerable attention due to the various connections to the socalled network design ...
Optimal BiLevel Augmentation for Selectively Enhancing Graph Connectivity with Applications
 in Proc. 2nd International Symp. on Computing and Combinatorics, vol. LNCS #1090
, 1996
"... Our main problem is abstracted from several optimization problems for protecting information in cross tabulated tables and for improving the reliability of communication networks. Given an undirected graph G and two vertex subsets H 1 and H 2 , the smallest bilevel augmentation problem is that of a ..."
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Cited by 1 (1 self)
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Our main problem is abstracted from several optimization problems for protecting information in cross tabulated tables and for improving the reliability of communication networks. Given an undirected graph G and two vertex subsets H 1 and H 2 , the smallest bilevel augmentation problem is that of adding to G the smallest number of edges such that G contains two internally vertexdisjoint paths between every pair of vertices in H 1 and two edgedisjoint paths between every pair of vertices in H 2 . We give a data structure to represent essential connectivity information of H 1 and H 2 simultaneously. Using this data structure, we solve the bilevel augmentation problem in O(n + m) time, where n and m are the numbers of vertices and edges in G. Our algorithm can be parallelized to run in O(log 2 n) time using n +m processors on an EREW PRAM. By properly setting G, H 1 and H 2 , our augmentation algorithm also subsumes several existing optimal algorithms for graph augmentation. 1 Int...