  Given a graph $G=(V,E)$, a function $f:V\rightarrow \{0,1,2\}$ is a total
Roman $\{2\}$-dominating function if: (1) every vertex $v\in V$ for which
$f(v)=0$ satisfies that $\sum_{u\in N(v)}f(u)\geq 2$, where $N(v)$ represents
the open neighborhood of $v$, and (2) every vertex $x\in V$ for which $f(x)\geq
1$ is adjacent to at least one vertex $y\in V$ such that $f(y)\geq 1$. The
weight of the function $f$ is defined as $\omega(f)=\sum_{v\in V}f(v)$. The
total Roman $\{2\}$-domination number, denoted by $\gamma_{t\{R2\}}(G)$, is the
minimum weight among all total Roman $\{2\}$-dominating functions on $G$. In
this article we introduce the concepts above and begin the study of its
combinatorial and computational properties. For instance, we give several
closed relationships between this parameter and other domination related
parameters in graphs. In addition, we prove that the complexity of computing
the value $\gamma_{t\{R2\}}(G)$ is NP-hard, even when restricted to bipartite
or chordal graphs.
