Overlapping cluster planarity

Cluster planarity is currently recognized as one of the most interesting problem in graph drawing. This paper investigates a new direction in this area by addressing the following question: let G be a graph along with a hierarchy of vertex clusters, where clusters can partially intersect. Does G admit a drawing where each cluster is inside a simple closed region, no two edges intersect, and no edge intersects a region twice? We investigate the interplay between this problem and the classical cluster planarity testing problem where clusters are not allowed to partially intersect. Characterizations, models, and algorithms are discussed.