A Roman dominating function (RDF) of a graph G is a labeling \(f:V(G)\longrightarrow \{0,1,2\}\) such that every vertex with label 0 has a neighbor with label 2. The weight of an RDF is the sum of its functions values over all vertices, and the Roman domination number of G is the minimum weight of an RDF of G. The Roman domination subdivision number \(\mathrm {sd}_{\gamma _{R}}(G)\) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the Roman domination number of G. In this paper, we present a new upper bound on the Roman domination subdivision number by showing that for every connected graph G of order at least three,$$\begin{aligned} \mathrm {sd}_{\gamma _{R}}(G)\le 3+\min \{\deg _2(v)\mid v\in V\;\mathrm {and} \;d(v)\ge 2\}, \end{aligned}$$where \(\deg _2(v)\) is the number of vertices of G at distance 2 from vertex v. Moreover, we show that the decision problem associated with \(\mathrm {sd}_{\gamma _{R}}(G)\) is NP-hard for bipartite graphs.

中文翻译：

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在图的罗马统治细分数上

图G的罗马支配函数（RDF）是标号\（f：V（G）\ longrightarrow \ {0,1,2 \} \），因此每个标号为0的顶点都有一个标号为2的邻居。重量的RDF的是其函数的值超过所有顶点的总和，和罗马控制数ģ是的RDF的最小重量ģ。罗马统治细分数\（\ mathrm {sd} _ {\ gamma _ {R}}（G）\）是必须细分的最小边数（G中的每个边最多可以细分一次）增加G的罗马统治次数。在本文中，我们通过显示对于每个阶数的连通图G至少三，$$ \ begin {aligned} \ mathrm {sd} _ {\ gamma _ {R}，提出了罗马统治细分数的新上限}（G）\ le 3+ \ min \ {\ deg _2（v）\ mid v \ in V \; \ mathrm {and} \; d（v）\ ge 2 \}，\ end {aligned} $$其中\（\ deg _2（v）\）是距顶点v距离2处G的顶点数量。此外，我们证明与\（\ mathrm {sd} _ {\ gamma _ {R}}（G）\）相关的决策问题对于二部图是NP-难的。