A connected graph G is said to be arbitrarily partitionable (AP for short) if for every partition ($n_1, ..., n_p$) of |V(G)| there exists a partition ($V_1, ..., V_p$) of V(G) such that each V_i induces a connected subgraph of G on $n_i$ vertices. Some stronger versions of this property were introduced, namely the ones of being online arbitrarily partitionable and recursively arbitrarily partitionable (OL-AP and R-AP for short, respectively), in which the subgraphs induced by a partition of G must not only be connected but also fulfil additional conditions. In this paper, we point out some structural properties of OL-AP and R-AP graphs with connectivity 2. In particular, we show that deleting a cut pair of these graphs results in a graph with a bounded number of components, some of whom have a small number of vertices. We obtain these results by studying a simple class of 2-connected graphs called balloons.