A total Roman dominating function on a graph $G$ is a function $% f:V(G)\rightarrow \{0,1,2\}$ such that every vertex $v$ with $f(v)=0$ is adjacent to some vertex $u$ with $f(u)=2$, and the subgraph of $G$ induced by the set of all vertices $w$ such that $f(w)>0$ has no isolated vertices. The weight of $f$ is $��_{v\in V(G)}f(v)$. The total Roman domination number $��_{tR}(G)$ is the minimum weight of a total Roman dominating function on $G$. A graph $G$ is $k$-$��_{tR}$-edge-critical if $��_{tR}(G+e)

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