Purely infinite simple reduced -algebras of one-relator separated graphs

Given a separated graph ( E , C ) , there are two different C ∗ -algebras associated to it: the full graph C ∗ -algebra C ∗ ( E , C ) and the reduced one C red ∗ ( E , C ) . For a large class of separated graphs ( E , C ) , we prove that C red ∗ ( E , C ) either is purely infinite simple or admits a faithful tracial state. The main tool we use to show pure infiniteness of reduced graph C ∗ -algebras is a generalization to the amalgamated case of a result on purely infinite simple free products due to Dykema.