Connectivity keeping trees in 2-connected graphs

摘要

Mader conjectured that for every positive integer k and finite tree T, every k-connected finite graph G with minimum degree delta(G) >= [3k/2] + vertical bar T vertical bar - 1 contains a subgraph T' congruent to 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 delta(G) >= 2 vertical bar T vertical bar - 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. (C) 2019 Elsevier B.V. All rights reserved.