关于拟k-连通图的一个注释

G是一个k-连通图，T是G的一个k-点割，若G-T可被划分成两个子图G₁，G₂，且|G₁|≥2，|G₂|≥2，则称T是G的一个非平凡点割。假定G是一个不含非平凡（k-1）点割的（k-1）-连通图，则称G是一个拟k-连通图。证明了对任意一个k≥5且t> $ \frac{k}{2}$的整数，若G是一个不含（K₂+tK₁）的k-连通图，且G中任意两个不同点对v，w，有d（v）+d（w）≥ $\frac{{3k}}{2} $+t，则对G中的任意一个点，存在一条与之关联的边收缩后可以得到一个拟k-连通图，且G中至少有$\frac{{\left| {V\left( G \right)} \right|}}{2} $条边使得收缩其中任意一条边后仍是拟k-连通的。

A note on quasi k-connected graphs

Let G be a k-connected graph,and T be a k-vertex-cut of a k-connected graph G.If G-T can be partitioned into subgraphs G₁ and G₂ such that|G₁|≥2,|G₂|≥2,then we call T a nontrivial k-vertex-cut of G.Suppose that G is a(k-1)-connected graph without nontrivial(k-1)-vertex-cut,then we call G a quasi k-connected graph.In this paper,we prove that for any integer k≥5 and t>k/2,if G is a(K₂+tK₁)-free k-connected graph for which d(v)+d(w)≥3 k/2+t for any pair v,w of distinct vertices of G,then every vertex of G is incident with an edge whose contraction yields a quasi k-connected graph,and so there are at least|V(G)|/2 edges of G such that the contraction of every member of the results in a quasi k-connected graph.