  Given a graph $G$, we say that an orientation $D$ of $G$ is a KT orientation
if, for all $u, v \in V(D)$, there is at most one directed path (in any
direction) between $u$ and $v$. Graphs that admit such orientations have been
used by Kierstead and Trotter (1992), Carbonero, Hompe, Moore, and Spirkl
(2023), Bria\'nski, Davies, and Walczak (2024), and Gir\~ao, Illingworth,
Powierski, Savery, Scott, Tamitegami, and Tan (2024) to construct graphs with
large chromatic number and small clique number that served as counterexamples
to various conjectures. Motivated by this, we consider which graphs admit KT
orientations (named after Kierstead and Trotter). In particular, we construct a
graph family with small independence number (sublinear in the number of
vertices) which admits a KT orientation. We show that the problem of
determining whether a given graph admits a KT orientation is NP-complete, even
if we restrict ourselves to planar graphs. Finally, we provide an algorithm to
decide if a graph with maximum degree at most 3 admits a KT orientation,
whereas, for graphs with maximum degree 4, the problem remains NP-complete.
