Connectivity keeping caterpillars and spiders in 2-connected graphs

Mader (2010) conjectured that for any tree T of order m, every k-connected graph G with minimum degree at least [Formula presented] contains a subtree T^′≅T such that G−V(T^′) is k-connected. A caterpillar is a tree in which a single path is incident to every edge. The conjecture has been proved when k=1 and for some special caterpillars when k=2. A spider is a tree with at most one vertex with degree more than 2. In this paper, we confirm the conjecture for all caterpillars and spiders when k=2.