Planarity Testing for C-Connected Clustered Graphs

We present a linear time algorithm for testing clustered planarity of c-connected clustered graphs and for computing a clustered planar embedding for such graphs. Our algorithm uses a decomposition of the input graph based on SPQR-trees and is the first linear time algorithm for clustered planarity testing. We define a normal form of clustered embeddings and show that a clustered graph is clustered planar if and only if any of its normal form embeddings is clustered planar. We also give a combinatorial characterization of clustered planar embeddings and show how to test clustered planarity of a given embedding of a clustered graph.