Total Roman domination subdivision number in graphs

A {\em Roman dominating function} on a graph $G$ is a function $f:V(G)\rightarrow \{0,1,2\}$ satisfying the condition that every vertex $u$ for which $f(u)=0$ is adjacent to at least one vertex $v$ for which $f(v)=2$. A {\em total Roman dominating function} is a Roman dominating function with the additional property that the subgraph of $G$ induced by the set of all vertices of positive weight has no isolated vertices. The weight of a total Roman dominating function $f$ is the value $\Sigma_{u\in V(G)}f(u)$. The {\em total Roman domination number} of $G$, $\gamma_{tR}(G)$, is the minimum weight of a total Roman dominating function on $G$. The {\em total Roman domination subdivision number} ${\rm sd}_{\gamma_{tR}}(G)$ of a graph $G$ 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 total Roman domination number. In this paper, we initiate the study of total Roman domination subdivision number in graphs and we present sharp bounds for this parameter.