How to Use Random Circulations to Find Small Cuts

Let K be an abelian group and G be a connected graph, both finite. Using basic properties of circulations, we show that it is easy to generate uniformly random K-circulations on G. This leads to efficient algorithms for computing the cut edges, cut edge-pairs, and cut vertices of a graph; for example, the cut edges are “usually” the edges where a random circulation vanishes. In the distributed setting, we improve the best known time complexity of any algorithm for finding cut edge-pairs to O(Diam), and for cut vertices to O(Diam +∆/ log |V |), where Diam is the diameter of the graph and ∆ is the maximum degree. Our algorithms are the Las Vegas kind and use messages of length O(log |V |). The distributed cut vertex algorithm can also be used to find the blocks of G.