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 $\lambda (v,w)=\min \{d(v),d(w)\}$. 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 $O({(n/\delta )}^2)$ cuts of size less than $\min \{\frac{3}{2}\lambda ,\delta \}$, and $O({(n/\delta )}^{\lfloor 2\alpha \rfloor })$ 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 $V(G)$ 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 $O(n/\delta )$ vertices and $O(n)$ edges.

我们提出了三个图的切割树，每个树都给出了对边缘连接结构的见解。所有三个割树的共同点是它们是相对于图的顶点集上给定的二元对称关系R来定义的，这概括了 Gomory-Hu 树。应用这些砍伐的树木，我们证明了以下内容：

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  一对顶点$\{v,w\}$图G是悬垂的，如果$\lambda （v,w）=\mathrm{分钟}\{d（v）,d（w）\}$。Mader 在 1974 年证明，每个具有最小度δ的简单图至少包含$\delta （\delta +1）/2$吊坠对。我们将这个下界改进为$\delta n/24$对于n 个顶点上的每个简单图G$\delta \ge 5$或者$\lambda \ge 4$或顶点连接$\kappa \ge 3$，并表明对于每个参数来说，在常数因子范围内这是最佳的。

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  每个简单图G都满足$\delta >0$有$氧（n/\delta ）$ δ -边连接的分量。此外，每个简单图G满足$0<\lambda <\delta$有$氧（{（n/\delta ）}^2）$切割尺寸小于$\mathrm{分钟}\{\frac{3}{2}\lambda ,\delta \}$， 和$氧（{（n/\delta ）}^{\lfloor 2\alpha \rfloor }）$最多切割尺寸$\mathrm{分钟}\{\alpha \cdot \lambda ,\delta -1\}$对于任何给定的实数$\alpha \ge 1$。

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  如果它或其补集在$V（G）$是一个单例。我们提供了 Lo 等人的以下最新结果的替代证明：给定一个在n 个顶点上的简单图G，满足$\delta >0$，我们可以在近线性时间内计算G的顶点子集，这样分别收缩这些顶点子集就可以保留G的所有非平凡最小割，并留下一个具有以下性质的图：$氧（n/\delta ）$顶点和$氧（n）$边缘。