**2. Preliminaries**

We start by providing some useful definitions and observations throughout the paper. **Definition 1.** *For any graph G, let*

$$\begin{array}{lcl}\mathcal{W}\_{\widetilde{\mathbb{G}}}^{\mathcal{R},1} &=& \{\boldsymbol{u} \in V \mid \text{there exists a } \gamma\_{\mathcal{R}}^{p}(\mathcal{G})\text{-function } f \text{ such that } f(\boldsymbol{u}) = 2\},\\\mathcal{W}\_{\widetilde{\mathbb{G}}}^{\mathcal{R},\leq 1} &=& \{\boldsymbol{u} \in V \mid f(\boldsymbol{u}) \leq 1 \text{ for some } \gamma\_{\mathcal{R}}^{p}(\mathcal{G})\text{-function } f\},\\\mathcal{W}\_{\widetilde{\mathbb{G}}}^{\mathcal{R},\geq 1} &=& \{\boldsymbol{u} \in V \mid \text{for each } \boldsymbol{v} \in \mathcal{N}\_{\mathcal{G}}(\boldsymbol{u}), f(\boldsymbol{v}) \leq 1 \text{ for every } \gamma\_{\mathcal{R}}^{p}(\mathcal{G})\text{-function } f\},\\\mathcal{W}\_{\widetilde{\mathbb{G}}}^{\mathcal{R},\boldsymbol{A}} &=& \{\boldsymbol{u} \in V \mid \boldsymbol{u} \text{ belongs to every } \gamma^{p}(\mathcal{G})\text{-set}\}.\end{array}$$

**Definition 2.** *Let u be a vertex of a graph G. A set S is said to be an almost perfect dominating set (almost PDS) with respect to u*, *(i) if each vertex x* ∈ *V* \ (*S* ∪ {*u*}) *has exactly one neighbor in S*, *and (ii) if u* ∈ *V* \ *S, then u has at most one neighbor in S. Let*

> *γ<sup>p</sup>*(*<sup>G</sup>*; *u*) = min{|*S*| : *S* is an almost PDS with respect to *<sup>u</sup>*}.

Trivially, every PDS of *G* is an almost PDS with respect to any vertex of *G* and thus *γ<sup>p</sup>*(*<sup>G</sup>*; *u*) is well defined. Hence *γ<sup>p</sup>*(*<sup>G</sup>*; *u*) ≤ *γ<sup>p</sup>*(*G*) for each vertex *u* ∈ *V*. Let

$$\mathcal{W}\_G^{APD} \quad = \quad \{ \mu \in V \mid \gamma^p(G; \mu) = \gamma^p(G) \}.$$

The proof of the following two results are given in [12].

**Observation 1.** *Let G be a graph.*


*3. For any leaf u of G, there is a γpR*(*G*)*-function f such that f*(*u*) ≤ 1*.*

**Proposition 1.** *Let G be a graph. G has a γpR*(*G*)*-function that assigns 2 to every end strong support vertex. Thus every end strong support vertex of a graph G belongs to WR*,<sup>1</sup> *G .*

The next result is a consequence of Observation 1 and Proposition 1.

**Corollary 1.** *Let u be an end strong support vertex of a graph H. If G is the graph obtained from H by adding a vertex x and an edge ux, then γ<sup>p</sup>*(*G*) = *γ<sup>p</sup>*(*H*) *and γpR*(*G*) = *<sup>γ</sup>pR*(*H*)*.*

**Proposition 2.** *Let H be a graph and u* ∈ *<sup>V</sup>*(*H*)*. If G is a graph obtained from H by adding a path P*2 : *x*1*x*2 *attached at u by an edge ux*1, *then:*

