On the structure of arbitrarily partitionable graphs with given connectivity

A graph G=(V,E) is arbitrarily partitionable if for any sequence τ of positive integers adding up to |V|, there is a sequence of vertex-disjoint subsets of V whose orders are given by τ, and which induce connected subgraphs. Such a graph models, e.g., a computer network which may be arbitrarily partitioned into connected subnetworks. In this paper we study the structure of such graphs and prove that unlike in some related problems, arbitrarily partitionable graphs may have arbitrarily many components after removing a cutset of a given size ≥2. The sizes of these components grow exponentially, though.