Mader's conjecture for graphs with small connectivity

Mader conjectured that for any tree $T$ of order $m$, every $k$-connected graph $G$ with minimum degree at least $\lfloor \frac{3k}{2}\rfloor +m-1$ contains a subtree $T^{\mathrm{\prime }}\cong T$ such that $G-V(T^{\mathrm{\prime }})$ is $k$-connected. In this paper, we give a characterization for a subgraph to contain an embedding of a specified tree avoiding some vertex. As a corollary, we confirm Mader’s conjecture for $k\le 3$.