  We start with a Gromov-hyperbolic surface bundle $E$ over a graph, and drill
out essential simple closed curves from fibers to obtain a drilled bundle $F$.
We prove that for such drilled bundles $F$, the fundamental group $\pi_1(F)$ is
relatively hyperbolic with $(\mathbb{Z}\oplus \mathbb{Z})$ peripheral groups.
Combining the relative hyperbolicity of $\pi_1(F)$ thus obtained with a theorem
of Wise, we establish virtually special cubulability of $\pi_1(F)$ provided
that the maximal undrilled subbundles of $F$ are cubulable.
