In which cases a fiber bundle over a circle is a graph-manifold?

Question

A fiber bundle over a circle $M^{3} \longrightarrow S^{1}$ with fiber a surface $F_{g}$ is characterized via a homeomorphism $\varphi \colon F_{g} \to F_{g}$. It can be one of the following: periodic, reducible or psevdo-anosov. In which cases $M^{3}$ is a graph-manifold?

Answer 1

$M$ is a graph manifold if and only if $\phi$ is not pseudo-Anosov and, in the reducible case, no $\phi$-orbit of components of the complete reduction of $\phi$ is pseudo-Anosov.

To prove this cut $M$ along the torus system $T$ obtained by suspending the canonical reducing system for $\phi$. If no component of the complete reduction of $\phi$ is pseudo-Anosov then each component of $M-T$ is Seifert fibered. If some $\phi$-orbit of components of the complete reduction is pseudo-Anosov then the corresponding component of $M-T$ is hyperbolic.