Connectivity preserving trees in k‐connected or k‐edge‐connected graphs

We show that for k≤2 $k\le 2$ and any tree T $T$ of order m $m$, every k $k$‐connected (respectively, k $k$‐edge‐connected) graph G $G$ with minimum degree at least max{Δ(T)+k,m−1} $\max \{{\rm{\Delta }}(T)+k,m-1\}$ contains a subtree T′≅T $T^{\prime} \cong T$ such that G−E(T′) $G-E(T^{\prime} )$ is k $k$‐connected (respectively, k $k$‐edge‐connected), where Δ(T) ${\rm{\Delta }}(T)$ denotes the maximum degree of T $T$. We also show that for any k≥1 $k\ge 1$ and any tree T $T$ of order m≥4 $m\ge 4$, every k $k$‐connected (respectively, k $k$‐edge‐connected) graph G $G$ of order n $n$ with minimum degree at least 2(k+m−p) $2(k+m-p)$ where p=k(k+1)+(m−1)(m−4)2n+2 $p=\unicode{x02308}\frac{k(k+1)+(m-1)(m-4)}{2n}\unicode{x02309}+2$ contains a subtree T′≅T $T^{\prime} \cong T$ such that G−E(T′) $G-E(T^{\prime} )$ is k $k$‐connected (respectively, k $k$‐edge‐connected). The lower bound of 2(k+m−p) $2(k+m-p)$ on the minimum degree of G $G$ can be improved to k+m−1 $k+m-1$ under several restrictions or assumptions.