Total restrained Roman domination

Let $G$ be a graph with vertex set $V(G)$. A Roman dominating function  (RDF) on a graph $G$ is a function $f:V(G)longrightarrow{0,1,2}$ such that every vertex $v$  with $f(v)=0$ is adjacent to a vertex  $u$ with $f(u)=2$. If $f$ is an RDF on $G$, then let $V_i={vin V(G): f(v)=i}$ for $iin{0,1,2}$. An RDF $f$ is called a restrained (total) Roman dominating function if the subgraph induced by $V_0$  (induced by $V_1cup V_2$) has no isolated vertex. A total and restrained Roman dominating function is a total restrained Roman  dominating function.  The total restrained Roman domination number $gamma_{trR}(G)$ on a graph $G$ is the minimum weight of a total restrained Roman dominating function on the graph $G$. We initiate the study of total restrained Roman domination number and present several sharp bounds on $gamma_{trR}G)$. In addition, we determine this parameter for some classes of graphs.