  Unlike $\mathbb{R}^{3}$, the homogeneous spaces $\mathbb{E}(-1,\tau)$ have a
great variety of entire vertical minimal graphs. In this paper we explore
conditions which guarantees that a minimal surface in $\mathbb{E}(-1,\tau)$ is
such a graph. More specifically: we introduce the definition of a generalized
slab in $\mathbb{E}(-1,\tau)$ and prove that a properly immersed minimal
surface of finite topology inside such a slab region has multi-graph ends.
Moreover, when the surface is embedded, the ends are graphs. When the surface
is embedded and simply connected, it is an entire graph.
