We study the eternal dominating number and the m-eternal dominating number on digraphs. We generalize known results on graphs to digraphs. We also consider the problem "oriented (m-)eternal domination", consisting in finding an orientation of a graph that minimizes its eternal dominating number. We prove that computing the oriented eternal dominating number is NP-hard and characterize the graphs for which the oriented m-eternal dominating number is 2. We also study these two parameters on trees, cycles, complete graphs, complete bipartite graphs, trivially perfect graphs and different kinds of grids and products of graphs.

中文翻译：

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图的有向图和方向上的永恒支配集

我们研究了有向图上的永恒主数和m-永恒主数。我们将图上的已知结果推广到有向图。我们还考虑了“定向（m-）永恒支配”问题，包括找到最小化其永恒支配数的图的方向。我们证明了计算有向永恒支配数是 NP-hard 并刻画了有向 m-永恒支配数为 2 的图。 我们还在树、圈、完全图、完全二部图、平凡完美图上研究了这两个参数以及不同种类的网格和图形的产品。