Geodesic
Structural Properties of Recursively Partitionable Graphs with Connectivity 2
Discussiones Mathematicae. Graph Theory, Tome 37 (2017) no. 1, pp. 89-115

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A connected graph G is 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 Vi induces a connected subgraph of G on ni 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.
Keywords: online arbitrarily partitionable graph, recursively arbitrarily partitionable graph, graph with connectivity 2, balloon graph 
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Baudon, Olivier; Bensmail, Julien; Foucaud, Florent; Pilśniak, Monika. Structural Properties of Recursively Partitionable Graphs with Connectivity 2. Discussiones Mathematicae. Graph Theory, Tome 37 (2017) no. 1, pp. 89-115. http://geodesic.mathdoc.fr/item/DMGT_2017_37_1_a7/