  A quasi-total Roman dominating function on a graph $G=(V, E)$ is a function
$f : V \rightarrow \{0,1,2\}$ satisfying the following:
  - every vertex $u$ for which $f(u) = 0$ is adjacent to at least one vertex
$v$ for which $f(v) =2$, and
  - if $x$ is an isolated vertex in the subgraph induced by the set of vertices
labeled with 1 and 2, then $f(x)=1$.
  The weight of a quasi-total Roman dominating function is the value
$\omega(f)=f(V)=\sum_{u\in V} f(u)$. The minimum weight of a quasi-total Roman
dominating function on a graph $G$ is called the quasi-total Roman domination
number of $G$. We introduce the quasi-total Roman domination number of graphs
in this article, and begin the study of its combinatorial and computational
properties.
