A Roman \(\{2\}\)-dominating function (R2DF) \(f:V\longrightarrow \{0,1,2\}\) of a graph \(G=(V, E)\) has the property that for every vertex \(v\in V\) with \(f(v)=0\) either there is a vertex \(u\in N(v)\) with \(f(u)=2\) or there are two vertices \(x,y\in N(v)\) with \(f(x)=f(y)=1\). The weight of f is the sum \(f(V)=\sum _{v\in V}f (v)\). The minimum weight of an R2DF on G is the Roman \(\{2\}\)-domination number of G. In this paper, we first show that the associated decision problem for Roman \(\{2\}\)-domination is NP-complete even when restricted to planar graphs. Then, we give a linear algorithm that computes the Roman \(\{2\}\)-domination number of a given unicyclic graph.

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关于罗马$$ \ {2 \} $$ {2}-支配（意大利支配）的算法复杂性

图\（G =（V，E）\）的罗马\（\ {2 \} \）主导函数（R2DF）\（f：V \ longrightarrow \ {0,1,2 \} \）具有对于每个具有\（f（v）= 0 \）的顶点\（v \ in V \）都有一个顶点\（u \ in N（v）\）具有\（f（u）= 2 \）或两个顶点\（x，y \ in N（v）\）中的\（f（x）= f（y）= 1 \）。f的权重是和\（f（V）= \ sum _ {v \ in V} f（v）\）。一个R2DF对最小重量ģ是罗马\（\ {2 \} \） -domination的数目ģ。在本文中，我们首先证明了即使限制在平面图上，罗马\（\ {2 \} \）统治的相关决策问题也是NP完全的。然后，我们给出一个线性算法，用于计算给定单环图的罗马\（\ {2 \} \）-支配数。