**On a Relation between the Perfect Roman Domination and Perfect Domination Numbers of a Tree**

**Zehui Shao 1, Saeed Kosari 1,\*, Mustapha Chellali 2, Seyed Mahmoud Sheikholeslami 3 and Marzieh Soroudi 3**


Received: 23 April 2020; Accepted: 7 June 2020; Published: 12 June 2020

**Abstract:** A *dominating set* 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* if each vertex in *V* − *S* is adjacent to exactly one vertex in *S*. The minimum cardinality of a *perfect dominating set* is the perfect domination number *<sup>γ</sup><sup>p</sup>*(*G*). A function *f* : *V*(*G*) → {0, 1, 2} is a perfect Roman dominating function (PRDF) on *G* if every vertex *u* ∈ *V* for which *f*(*u*) = 0 is adjacent to exactly one vertex *v* for which *f*(*v*) = 2. The weight of a PRDF is the sum of its function values over all vertices, and the minimum weight of a PRDF of *G* is the perfect Roman domination number *<sup>γ</sup>pR*(*G*). In this paper, we prove that for any nontrivial tree *T*, *γpR*(*T*) ≥ *γ<sup>p</sup>*(*T*) + 1 and we characterize all trees attaining this bound.

**Keywords:** Roman domination number; perfect Roman domination number; tree
