∆k-Confluent and Ok-Confluent Graphs

In this paper we extend the concept of ∆-confluence to ∆k-confluence by allowing more generalized junctions, called ∆k-junctions. We present an algorithm for recognizing graphs that are ∆k-confluent. We then generalize ∆k-confluence to Ok-confluence by allowing non-intersecting chords within a junction, resulting in Ok-junctions. We present an algorithm for recognizing graphs that are Ok-confluent. Finally, we show that the clique problem can be solved in polynomial time for ∆k-confluent graphs.