Extracting maximal information about sets of minimum cuts
 Dan Gusfield,
 Dalit Naor
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There are two wellknown, elegant, compact, and efficiently computed representations of selected minimum edge cuts in a weighted, undirected graphG=(V, E) withn nodes andm edges: at one extreme, the GomoryHu cut tree [12] represents \(\left( {\begin{array}{*{20}c} n \\ 2 \\ \end{array} } \right)\) minimum cuts, one for each pair of nodes inG; at the other extreme, the PicardQueyranne DAG [24] represents all the minimum cuts between a single pair of nodes inG. The GH cut tree is constructed with onlyn−1 maxflow computations, and the PQ DAG is constructed with one maxflow computation, plusO(m) additional time. In this paper we show how to marry these two representations, getting the best features of both. We first show that we can construct all \(\left( {\begin{array}{*{20}c} n \\ 2 \\ \end{array} } \right)\) DAGs, one for each fixed pair of nodes, using onlyn−1 maxflow computations as in [12], plusO(m) time per DAG as in [24]. This speeds up the obvious approach by a factor ofn. We then apply this approach to an unweighted graphG, to find all the edgeconnectivity cuts inG, i.e., cuts with capacity equal to the connectivity ofG. Matula [22] gave a method to find one connectivity cut inO(nm) time; we show thatO(nm) time suffices to represent all connectivity cuts compactly, and to list all of them explicitly. This improves the previous best time bound ofO(n ^{2} m) [3] for listing the connectivity cuts. The connectivity cuts are central in network reliability calculations. We then show how to find all pairs of nodes that are separated by at least one connectivity cut inO(nm) time.
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 Title
 Extracting maximal information about sets of minimum cuts
 Journal

Algorithmica
Volume 10, Issue 1 , pp 6489
 Cover Date
 19930701
 DOI
 10.1007/BF01908632
 Print ISSN
 01784617
 Online ISSN
 14320541
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Minimum cut
 Connectivity cut
 Allpairs maxflows
 Cut tree
 Equivalent flow tree
 DAG
 Edgeconnectivity
 Industry Sectors
 Authors

 Dan Gusfield ^{(1)}
 Dalit Naor ^{(1)}
 Author Affiliations

 1. Computer Science Division, University of California, 95616, Davis, CA, USA