Homomorphically full graphs are those for which every homomorphic image is isomorphic to a subgraph. We extend the definition of homomorphically full to oriented graphs in two different ways. For the first of these, we show that homomorphically full oriented graphs arise as quasi-transitive orientations of homomorphically full graphs. This in turn yields an efficient recognition and construction algorithms for these homomorphically full oriented graphs. For the second one, we show that the related recognition problem is GI-hard, and that the problem of deciding if a graph admits a homomorphically full orientation is NP-complete. In doing so we show the problem of deciding if two given oriented cliques are isomorphic is GI-complete.

翻译：同态完全图是指其所有同态像均同构于某个子图的图。我们通过两种不同方式将"同态完全"的定义推广至有向图。针对第一种推广，我们证明了同态完全有向图可表现为同态完全图的拟传递定向。这一结论进而为这类同态完全有向图提供了高效识别与构造算法。针对第二种推广，我们证明了相关识别问题是GI困难的，并且判定一个图是否存在同态完全定向的问题属于NP完全问题。在此过程中，我们同时证明了判定两个给定有向团是否同构的问题属于GI完全问题。