Mader's conjecture for graphs with small connectivity

Mader conjectured that for any treeT$T$ of orderm$m$ , everyk$k$ ‐connected graphG$G$ with minimum degree at least⌊ 3 k 2 ⌋ + m − 1$\lfloor \frac{3k}{2}\rfloor +m-1$ contains a subtreeT ′ ≅ T$T^{\prime} \cong T$ such thatG − V ( T ′ )$G-V(T^{\prime} )$ isk$k$ ‐connected. In this article, 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 fork ≤ 3$k\le 3$ .