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

Given a separated graph (E,C)(E,C), there are two different C∗C∗-algebras associated to it: the full graph C∗C∗-algebra C∗(E,C)C∗(E,C) and the reduced one Cred∗(E,C). For a large class of separated graphs (E,C)(E,C), we prove that Cred∗(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∗C∗-algebras is a generalization to the amalgamated case of a result on purely infinite simple free products due to Dykema.