Mader在2010年和2012年分别提出猜想：（1）对所有的正整数k和树T，每个最小度δ(G)≥[3k/2]-1+|T|的k-点连通图G都包含一个同构于T的子树T'使得G-V(T')仍然是k-点连通的；（2）每个满足最小度δ(D)≥2k+m-1的k-点连通有向图D都存在阶为m的有向路P使得D-V(P)仍然是k-点连通的。本项目的研究主要围绕着这两个猜想展开。具体地来说，我们会从下面三个方向来研究这两个猜想：对比较小的k；对一些特定的（定向）树，如（定向）星图和（定向）双星图；对一些特定图类，如加围长条件和二部（有向）图。在这些研究基础上，我们还将探索在一般（有向）图中解决Mader猜想的方法。

In 2010 and 2012, Mader proposed two conjectures: (1) For every positive integer k and every finite tree T, every k-vertex-connected graph G with minimum degree δ(G)≥[3k/2]-1+|T| contains a subtree T' isomorphic to T such that G-V(T') is still k-vertex-connected; (2) Every k-vertex-connected digraph D with minimum degree δ(D)≥2k+m-1 for a positive integer m has a dipath P with m vertices such that D-V(P) is still k-vertex-connected. In this project, we mainly focus on these two conjectures. Specifically, we will study these two conjectures in the following three directions: for the small k; for some (directed) trees, such as (directed) stars and (directed) double-stars; for some classes of graphs (digraphs), such as addding girth condition to graphs (digraphs) and the bipartite graphs (digraphs). Based on these studies, we will also explore methods to solve the two Mader's conjectures for general graphs (digraphs).