  In this paper, we study cut sets of attractors of iteration function systems
(IFS) in $\mathbb{R}^d$. Under natural conditions, we show that all irreducible
cut sets of these attractors are perfect sets or single points. This leads to a
criterion for the existence of cut points of IFS attractors. If the IFS
attractors are self-affine tiles, our results become algorithmically checkable
and can be used to exhibit cut points with the help of Hata graphs. This
enables us to construct cut points of some self-affine tiles studied in the
literature.
