Abstract Given a graph G , we consider γ r ( G ) , γ { R 2 } ( G ) , γ r 2 ( G ) and γ R ( G ) as the weak Roman domination number, the Roman { 2 } -domination number, the 2-rainbow domination number and the Roman domination number of G , respectively. It is known that γ r ( G ) ≤ γ { R 2 } ( G ) ≤ γ r 2 ( G ) ≤ γ R ( G ) holds for any graph G . In connection with this, constructive characterizations of the trees T that satisfy the equalities above that are related with the weak Roman domination number are given in this work. That is, the trees T for which γ r ( T ) = γ { R 2 } ( T ) , γ r ( T ) = γ r 2 ( T ) and γ r ( T ) = γ R ( T ) are described.

中文翻译：

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关于树中弱罗马统治的建设性特征

摘要 给定一个图 G ，我们考虑 γ r ( G ) , γ { R 2 } ( G ) , γ r 2 ( G ) 和 γ R ( G ) 作为弱罗马统治数，罗马 { 2 } -统治数，分别是 G 的 2-rainbow 支配数和罗马支配数。已知γ r ( G ) ≤ γ { R 2 } ( G ) ≤ γ r 2 ( G ) ≤ γ R ( G ) 对任何图G 都成立。与此相关，在这项工作中给出了满足上述与弱罗马统治数相关的等式的树 T 的建设性特征。即，描述了 γ r ( T ) = γ { R 2 } ( T ) 、 γ r ( T ) = γ r 2 ( T ) 和 γ r ( T ) = γ R ( T ) 的树 T。