We study the version of the C-Planarity problem in which edges connecting the same pair of clusters must be grouped into pipes, which generalizes the Strip Planarity problem. We give algorithms to decide several families of instances for the two variants in which the order of the pipes around each cluster is given as part of the input or can be chosen by the algorithm.
Angelini, P., Da Lozzo, G. (2019). Clustered Planarity with Pipes. ALGORITHMICA, 81(6), 2484-2526 [10.1007/s00453-018-00541-w].