Connectivity keeping trees in 2‐connected graphs
Toru Hasunuma, Kosuke Ono · Journal of Graph Theory · 2019
Abstract Mader [J Graph Theory 65 (2010), 61‐69] conjectured that for any tree of order , every ‐connected graph with minimum degree at least contains a subtree such that is ‐connected. In this paper, we show that for any tree of order , every 2‐connected graph with minimum degree at least contains a subtree such that is 2‐connected, where denotes the number of internal vertices of . Besides, the lower bound on the minimum degree can be improved to and if is a caterpillar and a quasi‐monotone caterpillar, respectively. From our results, it follows that Mader's conjecture for 2‐connected graphs is true for any tree with , any caterpillar with , or any quasi‐monotone caterpillar.