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 [3k/2]+ m-1 contains a subtree T' congruent to 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. Spider (C) 2020 Elsevier B.V. All rights reserved.