Confluent graphs capture the connection properties of train tracks, offering a very natural generalization of planar graphs, and -as the example of railroad maps shows - are an important tool in graph visualization. In this paper we continue the study of confluent graphs, introducing strongly confluent graphs and tree-confluent graphs. We show that strongly confluent graphs can be recognized in NP (the complexity of recognizing confluent graphs remains open). We also give a natural elimination ordering characterization of tree-confluent graphs which shows that they form a subclass of the chordal bipartite graphs, and can be recognized in polynomial time.
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