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：同态完全图是指其每一个同态像都与某个子图同构的图。我们以两种不同方式将同态完全的定义扩展到有向图。针对第一种扩展，我们证明了同态完全有向图可作为同态完全图的拟传递定向出现。这进而为识别与构造这类同态完全有向图提供了高效算法。针对第二种扩展，我们证明了相关的识别问题是GI-hard的，并且判定一个图是否允许存在同态完全定向的问题是NP完全的。在此过程中，我们证明了判定两个给定有向团是否同构的问题是GI完全的。