  We construct a hereditary class of triangle-free graphs with unbounded
chromatic number, in which every non-trivial graph either contains a pair of
non-adjacent twins or has an edgeless vertex cutset of size at most two. This
answers in the negative a question of Chudnovsky, Penev, Scott, and Trotignon.
The class is the hereditary closure of a family of (triangle-free) twincut
graphs $G_1, G_2, \ldots$ such that $G_k$ has chromatic number $k$. We also
show that every twincut graph is edge-critical.
