Connectivity keeping trees in 2-connected graphs

Mader conjectured that for every positive integer k and finite tree T, every k-connected finite graph G with minimum degree δ(G)≥⌊3k2⌋+|T|−1 contains a subgraph T′≅T such that G−V(T′) remains k-connected. The conjecture has been proved for some special cases: T is a path; k=1; k=2 and T is a star, double star, path-star or path-double-star. In this paper, we show that the conjecture holds when k=2 and T is a tree with diameter at most 4 or T is a caterpillar tree with diameter 5. We also show that the minimum degree condition δ(G)≥2|T|−ℓ+1 suffices for k=2 and T is a tree with at least ℓ leaves and at least 3 vertices. Our result extends the results of Tian et al. for T isomorphic to star or double-star.