  Topological structure of minimal sets is studied for a dynamical system
$(E,F)$ given by a fibre-preserving, in general non-invertible, continuous
selfmap $F$ of a graph bundle $E$. These systems include, as a very particular
case, quasiperiodically forced circle homeomorphisms. Let $M$ be a minimal set
of $F$ with full projection onto the base space $B$ of the bundle. We show that
$M$ is nowhere dense or has nonempty interior depending on whether the set of
so called endpoints of $M$ is dense in $M$ or is empty. If $M$ is nowhere
dense, we prove that either a typical fibre of $M$ is a Cantor set, or there is
a positive integer $N$ such that a typical fibre of $M$ has cardinality $N$. If
$M$ has nonempty interior we prove that there is a positive integer $m$ such
that a typical fibre of $M$, in fact even each fibre of $M$ over a \emph{dense
open} set $\mathcal O \subseteq B$, is a disjoint union of $m$ circles.
Moreover, we show that each of the fibres of $M$ over $B\setminus \mathcal O$
is a union of circles properly containing a disjoint union of $m$ circles.
Surprisingly, some of the circles in such "non-typical" fibres of $M$ may
intersect. We also give sufficient conditions for $M$ to be a sub-bundle of
$E$.
