**1. Introduction**

Domination theory is a classical and interesting topic in theory of graphs, as well as one of the most active areas of research in this topic. The increasing interest in this area is partly explained by the diversity of applications to both theoretical and real-world problems, such as facility location problems, monitoring communication, coding theory, algorithm design, complex ecosystems, electrical networks, among others. A set *D* ⊆ *V*(*G*) of vertices of a graph *G* is a dominating set if every vertex in *V*(*G*) \ *D* is adjacent to at least one vertex in *D*. The domination number of *G*, denoted by *<sup>γ</sup>*(*G*), is the minimum cardinality among all dominating sets of *G*. Many variants of the previous concept have appeared in the literature. We refer to [1,2] for numerous results on this issue.

A remarkable variant of the parameter above, and one of the most studied, is as follows. A dominating set *D* of a graph *G* without isolated vertices is a total dominating set if the subgraph induced by the vertices of *D* has no isolated vertex. Notice that any graph with no isolated vertex has a total dominating set, since *D* = *V*(*G*) is such a set. The total domination number of *G*, denoted by *<sup>γ</sup>t*(*G*), is the minimum cardinality among all total dominating sets of *G*. More information on total domination in graphs can be found in the survey [3] and the book [4].

Next, we consider another variant of the concept of domination. A semitotal dominating set of a graph *G* without isolated vertices, is a dominating set *D* of *G* such that every vertex in *D* is within distance two of another vertex of *D*. The semitotal domination number, denoted by *<sup>γ</sup>t*<sup>2</sup>(*G*), is the minimum cardinality among all semitotal dominating sets of *G*. This parameter was introduced by Goddard et al. in [5], and was also further studied in [6–8].

For any graph without isolated vertices, we have that every semitotal dominating set is also a dominating set. Similarly, every total dominating set is a semitotal dominating set. Hence, the next inequality chain, given in [5], relates the parameters above.

$$
\gamma(G) \le \gamma\_{t2}(G) \le \gamma\_t(G) \tag{1}
$$

In the last decades, functions defined on graphs have received much attention in domination theory. This fact may be because the classical (total) domination problem can be studied using functions defined on graphs. Based on this approach, we consider the following concepts, which are also variants of the domination in graphs.

Let *f* : *V*(*G*) → {0, 1, 2} be a function on a graph *G*. Notice that *f* generates three sets *V*0, *V*1 and *V*2, where *Vi* = {*v* ∈ *<sup>V</sup>*(*G*): *f*(*v*) = *i*} for *i* = 0, 1, 2. In this sense, from now on, we will write *f*(*V f*0 , *V f*1 , *V f*2 ) so as to refer to the function *f* . Given a set *S* ⊆ *<sup>V</sup>*(*G*), *f*(*S*) = ∑*v*∈*<sup>S</sup> f*(*v*). We define the *weight* of *f* as *ω*(*f*) = *f*(*V*(*G*)) = |*V f*1 | + 2|*V f*2 |. In this sense, by an *f*(*V*(*G*))-function, we mean a function of weight *f*(*V*(*G*)). If the function *f* is clear from the context, then we will simply write *f*(*<sup>V</sup>*0, *V*1, *<sup>V</sup>*2). We shall also use the following notations: *V*1,2 = {*v* ∈ *V*1 : *<sup>N</sup>*(*v*) ∩ *V*2 = ∅} and *V*1,1 = *V*1 \ *V*1,2.

Roman domination in graphs was formally defined by Cockayne, Dreyer, Hedetniemi, and Hedetniemi [9] motivated, in part, by an article in Scientific American of Ian Stewart entitled "Defend the Roman Empire" [10]. A Roman dominating function (RDF) on a graph *G* is a function *f*(*<sup>V</sup>*0, *V*1, *<sup>V</sup>*2) satisfying that every vertex *u* ∈ *V*0 is adjacent to at least one vertex *v* ∈ *V*2. The Roman domination number of *G*, denoted by *<sup>γ</sup>R*(*G*), is the minimum weight among all RDFs on *G*. Further results on Roman domination can be found for example, in [11–14].

Another kind of functions defined on graphs are the total Roman dominating functions, which were introduced by Liu and Chang [15] and later, studied by Abdollahzadeh Ahangar et al. in [16]. A total Roman dominating function (TRDF) on a graph *G* without isolated vertices, is an RDF *f*(*<sup>V</sup>*0, *V*1, *<sup>V</sup>*2) such that the set *V*1 ∪ *V*2 is a total dominating set of *G*. The minimum weight among all TRDFs on *G* is the total Roman domination number of *G* and it is denoted by *<sup>γ</sup>tR*(*G*).

Abdollahzadeh Ahangar et al. [16] give the next relationship between the total Roman domination number and the domination number of a graph: If *G* is a graph with no isolated vertex, then

$$2\gamma(G) \le \gamma\_{t\mathbb{R}}(G) \le 3\gamma(G). \tag{2}$$

Also, the authors of [16] proposed open problems concerning characterizing the graphs that satisfy the equalities in the inequality chain above. While the families of trees which satisfy these equalities has been characterized in [17], it remains an open problem to characterize graphs in general. In that sense, in this article we study the open problems above. In the next section we first give new lower and upper bounds for this parameter, which improve the bounds given in the Inequality chain (2). Also, in Section 3 we give a characterization for the graphs *G* that satisfy the equality *<sup>γ</sup>tR*(*G*) = <sup>2</sup>*γ*(*G*); and finally, in Section 4 we give some necessary conditions that satisfy the graphs *G* for which *<sup>γ</sup>tR*(*G*) = <sup>3</sup>*γ*(*G*).
