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A graph \(G = (V;E)\) is arbitrarily vertex decomposable if for any sequence \(\tau\) of positive integers adding up to \(|V|\), there is a sequence of vertex-disjoint subsets of \(V\) whose orders are given by \(\tau\), and which induce connected graphs. The main aim of this paper is to study the recursive version of this problem. We present a solution for trees, suns, and partially for a class of 2-connected graphs called balloons.