  In [W. Mader, Connectivity keeping paths in $k$-connected graphs, J. Graph
Theory 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, Discrete
Math. 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-star
and $k=2$.
