Abstract

In this report we explore some interesting combinatorial properties of minimum cuts of a graph such as submodularity of cuts and crossing cuts lemma. Our main goal is to represent all the (at most n(n-1)/2) min-cuts of a graph compactly, and retrieve them in O(k) time, where k is the size of any min-cut. We discuss various properties of the cactus, and the algorithm by Karzanov and Timofeev to construct the cactus of an unweighted, undirected graph in O(kn^2) time. Here we propose a new algorithm, which constructs the cactus of an weighted, undirected graph in O(nm + n^2 log n + tm log n) time, where t is the number of cycles in the resulting cactus.

Citations