In [W. Mader, Connectivity keeping paths in $k$-connected graphs, J. GraphTheory 65 (2010) 61-69.], Mader conjectured that for every positive integer $k$and every finite tree $T$ with order $m$, every $k$-connected, finite graph $G$with $\delta(G)\geq \lfloor\frac{3}{2}k\rfloor+m-1$ contains a subtree $T'$isomorphic to $T$ such that $G-V(T')$ is $k$-connected. In the same paper,Mader proved that the conjecture is true when $T$ is a path. Diwan and Tholiya[A.A. Diwan, N.P. Tholiya, Non-separating trees in connected graphs, DiscreteMath. 309 (2009) 5235-5237.] verified the conjecture when $k=1$. In this paper,we will prove that Mader's conjecture is true when $T$ is a star or double-starand $k=2$.
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机译：在[W. Mader，《连通性在连接$ k $的图中保持路径》，J。GraphTheory 65（2010）61-69。]，Mader猜想对于每个正整数$ k $和每个阶为$ m $的有限树$ T $，每个与$ k $连接的有限图$ G $与$ \ delta（G）\ geq \ lfloor \ frac {3} {2} k \ rfloor + m-1 $包含与$ T $同构的子树$ T'$这样，$ GV（T'）$连接了$ k $。在同一篇论文中，Mader证明了当$ T $是路径时，猜想是正确的。 Diwan和Tholiya [A.A。 N.P. Diwan Tholiya，连通图中的非分离树，DiscreteMath。 [309（2009）5235-5237。]验证了$ k = 1 $时的猜想。在本文中，我们将证明当$ T $是星或双星且$ k = 2 $时，Mader的猜想是正确的。
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