  A Roman $\{2\}$-dominating function (R2F) is a function $f:V\rightarrow
\{0,1,2\}$ with the property that for every vertex $v\in V$ with $f(v)=0$ there
is a neighbor $u$ of $v$ with $f(u)=2$, or there are two neighbors $x,y$ of $v$
with $f(x)=f(y)=1$. A total Roman $\{2\}$-dominating function (TR2DF) is an R2F
$f$ such that the set of vertices with $f(v)>0$ induce a subgraph with no
isolated vertices. The weight of a TR2DF is the sum of its function values over
all vertices, and the minimum weight of a TR2DF of $G$ is the total Roman
$\{2\}$-domination number $\gamma_{tR2}(G).$ In this paper, we initiate the
study of total Roman $\{2\}$-dominating functions, where properties are
established. Moreover, we present various bounds on the total Roman
$\{2\}$-domination number. We also show that the decision problem associated
with $\gamma_{tR2}(G)$ is NP-complete for bipartite and chordal graphs.
{Moreover, we show that it is possible to compute this parameter in linear time
for bounded clique-width graphs (including tres).}
