Fast Algorithms for Even/Odd Minimum Cuts and Generalizations
ESA '00 Proceedings of the 8th Annual European Symposium on Algorithms
Note: Flow trees for vertex-capacitated networks
Discrete Applied Mathematics
Flow equivalent trees in undirected node-edge-capacitated planar graphs
Information Processing Letters
An Õ(mn) Gomory-Hu tree construction algorithm for unweighted graphs
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Characterizing (quasi-)ultrametric finite spaces in terms of (directed) graphs
Discrete Applied Mathematics
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We show that there is no cut-tree for various connectivity concepts, hence pointing out to errors in the papers of Schnorr [SIAM J. Comput., 8 (1979), pp. 265-275] and Gusfield and Naor [Networks, 21 (1991), pp. 505-520]. Gomory and Hu [SIAM J. Appl. Math., 9 (1961), pp. 551-560] constructed a cut-tree for undirected graphs which compactly represents a minimum cut for each pair of vertices. This has a straightforward generalization to directed Eulerian graphs, cf. Gupta [SIAM J. Appl. Math., 15 (1967), pp. 168-171]. A generalization for arbitrary directed graphs was given by Schnorr. There is a well-known transformation of vetex connectivity to directed edge connectivity; directed edge cuts correspond to vertex cuts in some weak sense. The result of Schnorr was later applied by Gusfield and Naor [8] for such a cut-tree construction. In this paper counterexamples are described to show that for directed graphs there is no cut-tree and therefore the cut-tree results of Schnorr and Gusfield and Naor are incorrect. Our final example shows that, without weakening the notion of vertex connectivity, it is impossible to construct vertex cut-trees for undirected graphs in general.