A Roman dominating function of a graph $G=(V,E)$ is a labeling $f: V \rightarrow{} \{0 ,1, 2\}$ such that for each vertex $u \in V$ with $f(u) = 0$, there exists a vertex $v \in N(u)$ with $f(v) =2$. A Roman dominating function $f$ is a global Roman dominating function if it is a Roman dominating function for both $G$ and its complement $\overline{G}$. The weight of $f$ is the sum of $f(u)$ over all the vertices $u \in V$. The objective of Global Roman Domination problem is to find a global Roman dominating function with minimum weight. The objective of Global Roman Domination is to compute a global Roman dominating function of minimum weight. In this paper, we study the algorithmic aspects of Global Roman Domination problem on various graph classes and obtain the following results. 1. We prove that Roman domination and Global Roman Domination problems are not computationally equivalent by identifying graph classes on which one is linear-time solvable, while the other is NP-complete. 2. We show that Global Roman Domination problem is NP-complete on split graphs, thereby resolving an open question posed by Panda and Goyal [Discrete Applied Mathematics, 2023]. 3. We prove that Global Roman Domination problem is NP-complete on chordal bipartite graphs, planar bipartite graphs with maximum degree five and circle graphs. 4. On the positive side, we present a linear-time algorithm for Global Roman domination problem on cographs.

翻译：图 $G=(V,E)$ 的罗马控制函数是一个标记函数 $f: V \rightarrow{} \{0 ,1, 2\}$，使得对于每个满足 $f(u) = 0$ 的顶点 $u \in V$，都存在一个顶点 $v \in N(u)$ 满足 $f(v) =2$。若一个罗马控制函数 $f$ 同时是图 $G$ 及其补图 $\overline{G}$ 的罗马控制函数，则称 $f$ 为全局罗马控制函数。函数 $f$ 的权重定义为所有顶点 $u \in V$ 上 $f(u)$ 值的总和。全局罗马控制问题的目标是找到一个具有最小权重的全局罗马控制函数。本文研究了全局罗马控制问题在多种图类上的算法性质，并获得了以下结果：1. 我们通过识别出一些图类，证明罗马控制问题与全局罗马控制问题在计算复杂性上并不等价——在某些图类上，一个问题可在线性时间内求解，而另一个问题却是 NP 完全的。2. 我们证明了全局罗马控制问题在分裂图上是 NP 完全的，从而解决了 Panda 和 Goyal [Discrete Applied Mathematics, 2023] 提出的一个公开问题。3. 我们证明了全局罗马控制问题在弦二部图、最大度为五的平面二部图以及圆图上也是 NP 完全的。4. 从积极方面看，我们为全局罗马控制问题在补图（cographs）上提出了一个线性时间算法。