A tree T is arbitrarily vertex decomposable if for any sequence τ of positive integers adding up to the order of T there is a sequence of vertex-disjoint subtrees of T whose orders are given by τ; from a result by Barth and Fournier it follows that Δ (T) ≤ 4. A necessary and a sufficient condition for being an arbitrarily vertex decomposable star-like tree have been exhibited. The conditions seem to be very close to each other. © 2007 Elsevier B.V. All rights reserved.
Horňák, M., & Woźniak, M. (2008). On arbitrarily vertex decomposable trees. Discrete Mathematics, 308(7), 1268–1281. https://doi.org/10.1016/j.disc.2007.04.008