  We consider the problem of classifying those graphs that arise as an
undirected square of an oriented graph by generalising the notion of
quasi-transitive directed graphs to mixed graphs. We fully classify those
graphs of maximum degree three and those graphs of girth at least four that
arise an undirected square of an oriented graph. In contrast to the recognition
problem for graphs that admit a quasi-transitive orientation, we find it is
NP-complete to decide if a graph admits a partial orientation as a
quasi-transitive mixed graph. We prove the problem is Polynomial when
restricted to inputs of maximum degree three, but remains NP-complete when
restricted to inputs with maximum degree at least five. Our proof further
implies that for fixed $k \geq 3$, it is NP-complete to decide if a graph
arises as an undirected square of an orientation of a graph with $\Delta = k$.
