  In this paper, we give a necessary and sufficient condition for a graphical
strip in the Heisenberg group $\mathbb{H}$ to be area-minimizing in the slab
$\{-1<x<1\}$. We show that our condition is necessary by introducing a family
of deformations of graphical strips based on varying a vertical curve. We show
that it is sufficient by showing that strips satisfying the condition have
monotone epigraphs. We use this condition to show that any area-minimizing
ruled entire intrinsic graph in the Heisenberg group is a vertical plane and to
find a boundary curve that admits uncountably many fillings by area-minimizing
surfaces.
