图结构问题一直是图论的核心问题之一，连通图的可收缩子图与子式是图结构中的重要课题，它们对认清图的结构性质以及图论的应用有深刻的意义。令Z是具有某种性质的图的集合，G是Z中的图。若G存在子图H使得G/H还在Z中，则称H是G的Z-可收缩子图。连通图的可收缩子图和子式有密切的联系，决定一般图类的可收缩子图及其子式是一件困难的事情。本项目将从k-连通图出发寻找恰当的图类,使得在这个图类内阶较大的图都能够通过有限子图收缩的方式得到阶较小的图。刻画该图类内的可收缩子图以及子式极小的图的结构特征,以期能够反过来给出这类图的一种构造方法并由此为其他方面的研究提供一些有用的结构信息。具体而言，我们将先对k-连通图中可收缩子图存在的局部条件进行探索；再对k<8时拟k-连通图的拟k-可收缩子图进行研究，决定子式极小的图；最后将拟k-连通图推广，定义更一般的（k,h）-连通图并对（k,h）-连通图进行相关探索。

The structure theory of graphs is one of the key topics in graph theory. The contractible subgraphs and the graph minors are two important domains in the graph structure theory, they make sense for us to know the graph structure and the applications of graph theory..Let Q be a set of graphs with given properties. Let G be a graph of Q, a subgraph H of G is called Q-contractible if G/H is again a graph of Q. The Q-contractible subgraphs and the minors are closely related to each other. It is hard to tell whether a graph of Q has a Q-contractible subgraph. Also, it is hard to tell something about the graph which has no proper minor in Q.. In this project, we try to find a set of graphs Q, which contains all k-connected graphs as a subset such that, in Q, all graphs of large order has a Q-contractible subgraph with bounded order. We will try to figure out the conditions for the existence contractible subgraphs and its distribution, and try to characterize the properties of minor minimal graphs of Q. If things go well then we can generate all graphs of Q from some finite graphs, and can provide some useful structures information for the study of related domains. Specifically, we will discuss the local conditions for the existence k-contractible subgraph in k-connected graphs. For k<8, we will try to characterize the condition for the existence of quasi k-contractible subgraph, and its distribution. Further，we will try to determine the gap between a quasi k-connected graph and its minor. Finally, we will generalize the definition of quasi k-connected graphs to (k,h)-connected graphs and discuss some related questions of (k,h)-connected graphs.