Abstract
We study the number of inclusion-minimal cuts in an undirected connected graph G, also called \(st\)-cuts, for any two distinct nodes s and t: the \(st\)-cuts are in one-to-one correspondence with the partitions \(S \cup T\) of the nodes of G such that \(S \cap T = \emptyset \), \(s \in S\), \(t \in T\), and the subgraphs induced by S and T are connected. It is easy to find an exponential upper bound to the number of \(st\)-cuts (e.g. if G is a clique) and a constant lower bound. We prove that there is a more interesting lower bound on this number, namely, \(\varOmega (m)\), for undirected m-edge graphs that are biconnected or triconnected (2- or 3-node-connected). The wheel graphs show that this lower bound is the best possible asymptotically.
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Notes
- 1.
Since G is connected, also G[S] and G[T] are connected, otherwise we could remove at least one edge from the minimal cutset to reconnect G[S] or G[T].
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This work was partially supported by JST CREST, grant number JPMJCR1401, Japan, and MIUR, Italy.
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Conte, A., Grossi, R., Marino, A., Rizzi, R., Uno, T., Versari, L. (2018). Tight Lower Bounds for the Number of Inclusion-Minimal st-Cuts. In: Brandstädt, A., Köhler, E., Meer, K. (eds) Graph-Theoretic Concepts in Computer Science. WG 2018. Lecture Notes in Computer Science(), vol 11159. Springer, Cham. https://doi.org/10.1007/978-3-030-00256-5_9
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