Summary: W. Mader [J. Graph Theory 65, No. 1, 61–69 (2010; Zbl 1234.05145)] 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 \(T^\prime\) isomorphic to \(T\) such that \(G - V(T^\prime)\) 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 [loc. cit.] conjectured that every \(k\)-connected digraph \(D\) with minimum semi-degree \(\delta(D) = \min \{\delta^+(D), \delta^-(D) \} \geq 2 k + m - 1\) 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.