Nonseparating trees in 2-connected graphs and oriented trees in strongly connected digraphs

Mader [J. Graph Theory 65 (2010) 61-69] 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. The conjecture has been verified for paths, trees when $k=1$, and stars or double-stars when $k=2$. In this paper we verify the conjecture for two classes of trees when $k=2$. For digraphs, Mader [J. Graph Theory 69 (2012) 324-329] conjectured that every $k$-connected digraph $D$ with minimum semi-degree $\delta(D)=min\{\delta^+(D),\delta^-(D)\}\geq 2k+m-1$ for a positive integer $m$ has a dipath $P$ of order $m$ with $\kappa(D-V(P))\geq k$. The conjecture has only been verified for the dipath with $m=1$, and the dipath with $m=2$ and $k=1$. In this paper, we prove that every strongly connected digraph with minimum semi-degree $\delta(D)=min\{\delta^+(D),\delta^-(D)\}\geq m+1$ contains an oriented tree $T$ isomorphic to some given oriented stars or double-stars with order $m$ such that $D-V(T)$ is still strongly connected.