L-space intervals for graph manifolds and cables

We present a graph manifold analog of the Jankins–Neumann classification of Seifert fibered spaces over$S^{2}$admitting taut foliations, providing a finite recursive formula to compute the L-space Dehn-filling interval for any graph manifold with torus boundary. As an application of a generalization of this result to Floer simple manifolds, we compute the L-space interval for any cable of a Floer simple knot complement in a closed three-manifold in terms of the original L-space interval, recovering a result of Hedden and Hom as a special case.