  We study step skew-products over a finite-state shift (base) space whose
fiber maps are $C^1$ injective maps on the unit interval. We show that certain
invariant sets have a multi-graph structure and can be written graphs of one,
two or more functions defined on the base. In particular, this applies to any
hyperbolic set and to the support of any ergodic hyperbolic measure. Moreover,
within the class of step skew-products whose interval maps are 'absorbing',
open and densely the phase space decomposes into attracting and repelling
double-strips such that their attractors and repellers are graphs of one
single-valued or bi-valued continuous function almost everywhere, respectively.
