Let $G=(V,E)$ be a simple undirected graph. A Roman dominating function on $G$ is a function $f: V\to \{0,1,2\}$ satisfying the condition that every vertex $u$ with $f(u)=0$ is adjacent to at least one vertex $v$ with $f(v)=2$. The weight of a Roman dominating function is the value $f(G)=\sum_{u\in V} f(u)$. The Roman domination number of $G$ is the minimum weight of a Roman dominating function on $G$. The Roman bondage number of a nonempty graph $G$ is the minimum number of edges whose removal results in a graph with the Roman domination number larger than that of $G$. Rad and Volkmann [9] proposed a problem that is to determine the trees $T$ with Roman bondage number $1$. In this paper, we characterize trees with Roman bondage number $1$.

中文翻译：

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罗马束缚数为1的树木的特征

令$ G =（V，E）$是一个简单的无向图。$ G $上的罗马支配函数是函数$ f：V \ to \ {0,1,2 \} $满足以下条件：每个具有$ f（u）= 0 $的顶点$ u $至少相邻$ f（v）= 2 $的一个顶点$ v $。罗马支配函数的权重是值$ f（G）= \ sum_ {u \ in V} f（u）$。$ G $的罗马支配数是$ G $上罗马支配函数的最小权重。非空图$ G $的罗马束缚数是边的最小数量，这些边的去除会导致图的罗马支配数大于$ G $。Rad和Volkmann [9]提出了一个问题，即确定具有罗马束缚数$ 1 $的树木$ T $。在本文中，我们用罗马束缚数$ 1 $表征树木。