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 psevdoanosov. In which cases $M^{3}$ is a graphmanifold?
$M$ is a graph manifold if and only if $\phi$ is not pseudoAnosov and, in the reducible case, no $\phi$orbit of components of the complete reduction of $\phi$ is pseudoAnosov. 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 pseudoAnosov then each component of $MT$ is Seifert fibered. If some $\phi$orbit of components of the complete reduction is pseudoAnosov then the corresponding component of $MT$ is hyperbolic. 

