Generalized cut trees for edge-connectivity

We present three cut trees of graphs, each of them giving insights into the edge-connectivity structure. All three cut trees have in common that they are defined with respect to a given binary symmetric relation R on the vertex set of the graph, which generalizes Gomory-Hu trees. Applying these cut trees, we prove the following:A pair of vertices {v, w} of a graph G is pendant if lambda(v, w) = min{d(v), d(w)}. Mader showed in 1974 that every simple graph with minimum degree delta contains at least delta(delta + 1)/2 pendant pairs. We improve this lower bound to delta n/24 for every simple graph G on n vertices with delta > 5 or lambda >= 4 or vertex connectivity kappa >= 3, and show that this is optimal up to a constant factor with regard to every parameter.Every simple graph G satisfying delta > 0 has O(n/delta) delta-edge-connected components. Moreover, every simple graph G that satisfies 0 < lambda < delta has O((n/delta )(2)) cuts of size less than min{ 3/2 lambda, delta}, and O((n/delta )1(2 alpha)I) cuts of size at most min{alpha lambda, delta - 1} for any given real number alpha > 1. A cut is trivial if it or its complement in V(G) is a singleton. We provide an alternative proof of the following recent result of Lo et al.: Given a simple graph G on n vertices that satisfies delta > 0, we can compute vertex subsets of G in near-linear time such that contracting these vertex subsets separately preserves all non-trivial min-cuts of G and leaves a graph having O(n/5) vertices and O(n) edges.