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

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.