Tight Lower Bounds for the Number of Inclusion-Minimal st-Cuts

  • Alessio Conte
  • Roberto Grossi
  • Andrea Marino
  • Romeo Rizzi
  • Takeaki Uno
  • Luca VersariEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11159)


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.



This work was partially supported by JST CREST, grant number JPMJCR1401, Japan, and MIUR, Italy.


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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Alessio Conte
    • 1
  • Roberto Grossi
    • 2
  • Andrea Marino
    • 2
  • Romeo Rizzi
    • 3
  • Takeaki Uno
    • 1
  • Luca Versari
    • 2
    Email author
  1. 1.National Institute of InformaticsTokyoJapan
  2. 2.Università di PisaPisaItaly
  3. 3.Università di VeronaVeronaItaly

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