RECURSIVELY ARBITRARILY VERTEX-DECOMPOSABLE GRAPHS

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.