  Bisimulations are standard in modal logic and, more generally, in the theory
of state-transition systems. The quotient structure of a Kripke model with
respect to the bisimulation relation is called a bisimulation contraction. The
bisimulation contraction is a minimal model bisimilar to the original model,
and hence, for (image-)finite models, a minimal model modally equivalent to the
original. Similar definitions exist for bounded bisimulations
($k$-bisimulations) and bounded bisimulation contractions. Two finite models
are $k$-bisimilar if and only if they are modally equivalent up to modal depth
$k$. However, the quotient structure with respect to the $k$-bisimulation
relation does not guarantee a minimal model preserving modal equivalence to
depth $k$. In this paper, we remedy this asymmetry to standard bisimulations
and provide a novel definition of bounded contractions called rooted
$k$-contractions. We prove that rooted $k$-contractions preserve
$k$-bisimilarity and are minimal with this property. Finally, we show that
rooted $k$-contractions can be exponentially more succinct than standard
$k$-contractions.
