The nuclear dimension of graph C*-algebras

Consider a graph C⁎C⁎-algebra C⁎(E)C⁎(E) with a purely infinite ideal I (possibly all of C⁎(E)C⁎(E)) such that I has only finitely many ideals and C⁎(E)/IC⁎(E)/I is approximately finite dimensional. We prove that the nuclear dimension of C⁎(E)C⁎(E) is 1. If I has infinitely many ideals, then the nuclear dimension of C⁎(E)C⁎(E) is either 1 or 2.