Structural Properties of Recursively Partitionable Graphs with Connectivity 2

A connected graph Gis said to be arbitrarily partitionable(AP for short) if for every partition (n1, . . . , np) of |V(G)| there exists a partition (V1, . . . , Vp) of V(G) such that each Viinduces a connected subgraph of Gon nivertices. Some stronger versions of this property were introduced, namely the ones of being online arbitrarily partitionableand recursively arbitrarily partitionable(OL-AP and R-AP for short, respectively), in which the subgraphs induced by a partition of Gmust 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.