  We show that clustered planarity with overlapping clusters as introduced by
Didimo et al. can be solved in polynomial time if each cluster induces a
connected subgraph. It can be solved in linear time if the set of clusters is
the union of two partitions of the vertex set such that, for each cluster, both
the cluster and its complement, induce connected subgraphs. Clustered planarity
with overlapping clusters is NP-complete, even if restricted to instances where
the underlying graph is 2-connected, the set of clusters is the union of two
partitions and each cluster contains at most two connected components while
their complements contain at most three connected components.
