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 $\Delta = k$.

翻译：我们考虑将拟传递有向图的概念推广至混合图，从而分类那些作为有向图的无向平方出现的图的问题。我们完整分类了最大度为三的图以及围长至少为四的图中，可作为有向图无向平方出现的情况。与允许拟传递定向的图的识别问题相反，我们判定一个图是否允许作为拟传递混合图的部分定向问题是NP完全的。我们证明当输入限制为最大度不超过三时该问题是多项式可解的，但当输入限制为最大度至少为五时仍为NP完全的。我们的证明进一步表明，对于固定的 $k \geq 3$，判定一个图是否可作为最大度 $\Delta = k$ 的图定向后的无向平方出现是NP完全的。