Purely infinite simple reduced C*-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 $C^{\ast }(E,C)$, and the reduced one $C_{\text{red}}^{\ast }(E,C)$. For a large class of separated graphs $(E,C)$, we prove that $C_{\text{red}}^{\ast }(E,C)$ 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.