Generalized cut trees for edge-connectivity

Abstract

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:

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  A pair of vertices $\{v,w\}$ of a graph G is pendant if $\lambda (v,w)=\min \left\{d\right(v),d(w\left)\right\}$. Mader showed in 1974 that every simple graph with minimum degree δ 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 $\delta \ge 5$ or $\lambda \ge 4$ or vertex connectivity $\kappa \ge 3$, and show that this is optimal up to a constant factor with regard to every parameter.

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  Every simple graph G satisfying $\delta >0$ has $O(n/\delta )$ δ-edge-connected components. Moreover, every simple graph G that satisfies $0<\lambda <\delta$ has $O\left({(n/\delta )}^2\right)$ cuts of size less than $\min \{\frac{3}{2}\lambda ,\delta \}$, and $O\left({(n/\delta )}^{\lfloor 2\alpha \rfloor }\right)$ cuts of size at most $\min \{\alpha \cdot \lambda ,\delta -1\}$ for any given real number $\alpha \ge 1$.

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  A cut is trivial if it or its complement in $V\left(G\right)$ 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 $O(n/\delta )$ vertices and $O\left(n\right)$ edges.