**1. Introduction**

In this paper, only simple and undirected graph without isolated vertices will be considered. The set of vertices of the graph *G* is denoted by *V* = *V*(*G*) and the edge set is *E* = *<sup>E</sup>*(*G*). The order of a graph *G* is the number of vertices of the graph *G* and it is denoted by *n* = *<sup>n</sup>*(*G*). The size of *G* is the cardinality of the edge set and it is denoted by *m* = *<sup>m</sup>*(*G*). For a vertex *v* ∈ *V*, the *open neighbourhood <sup>N</sup>*(*v*) is the set {*u* ∈ *V*(Γ) : *uv* ∈ *<sup>E</sup>*(*G*)}, the *closed neighbourhood* of *v* is the set *<sup>N</sup>*[*v*] = *<sup>N</sup>*(*v*) ∪ {*v*}, and the *degree* of *v* is deg*G*(*u*) = |*N*(*v*)|. Any vertex of degree one is called a *leaf*, a *support vertex* is a vertex adjacent to a leaf, a *strong support vertex* is a support vertex adjacent to at least two leaves and an *end support vertex* is a support vertex such that all its neighbors, except possibly one, are leaves. For a graph *G*, let *L*(*G*) = {*v* ∈ *V*(*G*) | deg*G*(*v*) = 1} and *Lv* = *<sup>N</sup>*(*v*) ∩ *<sup>L</sup>*(*G*). The *distance dG*(*<sup>u</sup>*, *v*) between two vertices *u* and *v* in a connected graph *G* is the length of a shortest *u* − *v* path in *G*. The *diameter* of *G*, denoted by diam(*G*), is the maximum value among distances between all pair of vertices of *G*. For a vertex *v* in a rooted tree *T*, let *<sup>C</sup>*(*v*) and *<sup>D</sup>*(*v*) denote the set of children and descendants of *v*, respectively and let *<sup>D</sup>*[*v*] = *<sup>D</sup>*(*v*) ∪ {*v*}. Moreover, the depth of *v*, depth(*v*), is the largest distance from *v* to a vertex in *<sup>D</sup>*(*v*). The *maximal subtree rooted at v*, denoted by *Tv*, consists of *v* and all its descendants. We write *Pn* for the *path* of order *n*. A tree *T* is a *double star* if it contains exactly two vertices that are not leaves. A double star with, respectively *p* and *q* leaves attached at each support vertex is denoted *DSp*,*q*. For a real-valued function *f* : *V* −→ R, the weight of *f* is *w*(*f*) = ∑*v*∈*<sup>V</sup> f*(*v*), and for *S* ⊆ *V* we define *f*(*S*) = ∑*v*∈*<sup>S</sup> f*(*v*). So *w*(*f*) = *f*(*V*).

A *dominating set* (DS) in a graph *G* is a set of vertices *S* ⊆ *V*(*G*) such that any vertex of *V* − *S* is adjacent to at least one vertex of *S*. A *dominating set S* of *G* is said to be a *perfect dominating set* (PDS) if each vertex in *V* − *S* is adjacent to exactly one vertex in *S*. The minimum cardinality of a *(perfect) dominating set* of a graph *G* is the *(perfect) domination number γ*(*G*) (*γ<sup>p</sup>*(*G*)). Perfect domination was introduced by Livingston and Stout in [1] and has been studied by several authors [2–6].

A function *f* : *V*(Γ) → {0, 1, 2} is a *Roman dominating function* (RDF) on *G* if every vertex *u* ∈ *V* for which *f*(*u*) = 0 is adjacent to at least one vertex *v* for which *f*(*v*) = 2. A *perfect Roman dominating function* (PRDF) on a graph *G* is an RDF *f* such that every vertex assigned a 0 is adjacent to exactly one vertex assigned a 2 under *f* . The minimum weight of a (perfect) RDF on a graph *G* is the *(perfect) Roman domination number <sup>γ</sup>R*(*G*) (*γpR*(*G*)). A (perfect) RDF on *G* with weight *<sup>γ</sup>R*(*G*) (*γpR*(*G*)) is called a *<sup>γ</sup>R*(*G*)-function (*γpR*(*G*)-function). An RDF *f* on a graph *G* = (*<sup>V</sup>*, *E*) can be represented by the ordered partition (*<sup>V</sup>*0, *V*1, *<sup>V</sup>*2) of *V*, where *Vi* = {*v* ∈ *V*| *f*(*v*) = *i*} for *i* = 0, 1, 2. The concept of Roman domination was introduced by Cockayne et al. in [7] and was inspired by the manuscript of the authors of [8], and Stewart [9] about the defensive strategy of the Roman Empire decreed by Constantine I The Great, while perfect Roman domination was introduced by Henning, Klostermeyer and MacGillivray in [10] and has been studied in [11–13]. For more on Roman domination, we refer the reader to the book chapters [14,15] and surveys [16–18].

It was shown in [10] that for any tree *G* of order *n* ≥ 3, *γpR*(*G*) ≤ 4*n*5 . Moreover, the authors have characterized all trees attaining this upper bound. Note that the previous upper bound have been improved by Henning and Klostermeyer [13] for cubic graphs of order *n* by showing that *γpR*(*G*) ≤ 3*n*4 .

It is worth mentioning that if *S* is a minimum *(perfect) dominating set* of a graph *G*, then clearly (*V* − *S*, ∅, *S*) is a (perfect) RDF and thus

$$
\gamma\_R(G) \le 2\gamma(G) \quad \text{and} \quad \gamma\_R^p(G) \le 2\gamma^p(G). \tag{1}
$$

On the other hand, if *f* = (*<sup>V</sup>*0, *V*1, *<sup>V</sup>*2) is a *<sup>γ</sup>R*(*G*)-function, then *V*1 ∪ *V*2 is a *dominating set* of *G* yielding

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

It is natural to ask whether the inequality (2) remains valid between *γ<sup>P</sup>*(*G*) and *γpR*(*G*) for any graph *G*. The answer is negative as it can be seen by considering the graph *H* obtained from a double star *DSp*,*p*,(*<sup>p</sup>* ≥ 3) with central vertices *u*, *v* by subdividing the edge *uv* with vertex *w*, and adding 2*k* (*k* ≥ 3) new vertices, where *k* vertices are attached to both *u* and *w* and the remaining *k* vertices are attached to both *v* and *w* (see Figure 1). Clearly, *γ<sup>p</sup>*(*H*) = 2*k* + 3 while *γpR*(*H*) = 5 and so the difference *γ<sup>p</sup>*(*H*) − *γpR*(*H*) can be even very large.

**Figure 1.** The graph *H*.

Motivated by the above example, we shall show in this paper that *γpR*(*T*) ≥ *γ<sup>p</sup>*(*T*) + 1 for every nontrivial tree *T*, and we characterize all trees attaining this bound.
