Let G be a graph with no isolated vertex and $f:V\left(G\right)\to \{0,1,2\}$ a function. If f satisfies that every vertex in the set $\{v\in V(G):f(v)=0\}$ is adjacent to at least one vertex in the set $\{v\in V(G):f(v)=2\}$, and if the subgraph induced by the set $\{v\in V(G):f(v)\ge 1\}$ has no isolated vertex, then we say that f is a total Roman dominating function on G. The minimum weight $\omega \left(f\right)={\sum }_{v\in V\left(G\right)}f\left(v\right)$ among all total Roman dominating functions f on G is the total Roman domination number of G. In this article we study this parameter for the rooted product graphs. Specifically, we obtain closed formulas and tight bounds for the total Roman domination number of rooted product graphs in terms of domination invariants of the factor graphs involved in this product.

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根产品图的总罗马统治数

令G为无孤立顶点的图，$F：V（G）\to \{0，\mathrm{1个}，2\}$一个功能。如果f满足集合中的每个顶点$\{v\in V（G）：F（v）=0\}$ 与集合中的至少一个顶点相邻 $\{v\in V（G）：F（v）=2\}$，如果子图由集合引起 $\{v\in V（G）：F（v）\ge \mathrm{1个}\}$没有孤立的顶点，那么我们说f是G的总罗马支配函数。最小重量$\omega （F）={\sum }_{v\in V（G）}F（v）$所有总罗马支配功能之中˚F上ģ是总罗马控制数ģ。在本文中，我们将针对根产品图研究此参数。具体来说，我们根据该乘积涉及的因子图的支配不变性，为有根乘积图的总罗马支配数获得封闭式和严格边界。