**Proof.**

1. For a *γ<sup>p</sup>*(*H*)-set *S*, let *S* = *S* ∪ {*<sup>x</sup>*1} if *u* ∈ *S*, and *S* = *S* ∪ {*<sup>x</sup>*2} if *u* ∈ *S*. Clearly, *S* is a PDS of *G* and thus *γ<sup>p</sup>*(*G*) ≤ *γ<sup>p</sup>*(*H*) + 1.

Now let *f* be a *γpR*(*G*)-function. Obviously, *f*(*<sup>x</sup>*1) + *f*(*<sup>x</sup>*2) ≥ 1. If *f*(*u*) ≥ 1, then the function *f* restricted to *H* is a PRDF on *H* yielding *γpR*(*G*) ≥ *γpR*(*H*) + 1. Thus assume that *f*(*u*) = 0. Then *f*(*<sup>x</sup>*1) + *f*(*<sup>x</sup>*2) = 2 and the function *g* : *V*(*H*) → {0, 1, 2} defined by *g*(*u*) = 1 and *g*(*x*) = *f*(*x*) for *x* ∈ *V*(*H*) \ {*u*} is a PRDF on *H* of weight *γpR*(*G*) − 1. Hence in any case, *γpR*(*G*) ≥ *γpR*(*H*) + 1.


For a graph *G* and a vertex *u* of *G*, we denote by *GuK*1,3 the graph obtained from *G* by adding a star *<sup>K</sup>*1,3 and joining one of its leaf to *u*.

**Proposition 3.** *Let G be a graph and u a vertex of G.*


 *3. If u* ∈ *WR*,≤<sup>1</sup> *G , then <sup>γ</sup>pR*(*GuK*1,3) = *γpR*(*G*) + 2*.*

**Proof.** Let *x* be the center of the star *<sup>K</sup>*1,3 and *x*1 a leaf of *<sup>K</sup>*1,3 attached at *u* by an edge *ux*1.

1. For a *γ<sup>p</sup>*(*G*)-set *S*, let *S* = *S* ∪ {*<sup>x</sup>*, *<sup>x</sup>*1} if *u* ∈ *S*, and *S* = *S* ∪ {*x*} for otherwise. Clearly, *S* is a PDS of *GuK*1,3and thus *<sup>γ</sup><sup>p</sup>*(*GuK*1,3) ≤ *γ<sup>p</sup>*(*G*) + 2.

Now, let *f* be a *<sup>γ</sup>pR*(*GuK*1,3 )-function. By Proposition 1, we may assume that *f*(*x*) = 2. If *f*(*<sup>x</sup>*1) ≤ 1, then the function *f* restricted to *G* is a PRDF on *G* of weight at most *<sup>γ</sup>pR*(*GuK*1,3 ) − 2. Thus, we assume that *f*(*<sup>x</sup>*1) = 2. Then the function *g* : *V*(*G*) → {0, 1, 2} defined by *g*(*u*) = 1 and *g*(*x*) = *f*(*x*) for all *x* ∈ *V*(*G*) \ {*u*} is a PRDF on *G* of weight *<sup>γ</sup>pR*(*GuK*1,3 ) − 3. In any case, *γpR*(*G*) ≤ *<sup>γ</sup>pR*(*GuK*1,3) − 2.


**Proposition 4.** *Let G be a graph and let u be an end support vertex of G which is adjacent to a strong support vertex v. If G is a graph obtained from G by adding a vertex x and an edge ux, then γ<sup>p</sup>*(*G*) = *γ<sup>p</sup>*(*G*) *and γpR*(*G*) ≥ *<sup>γ</sup>pR*(*G*)*. Moreover, if u* ∈ *WR*,<sup>1</sup> *G , then γpR*(*G*) = *<sup>γ</sup>pR*(*G*)*.*

**Proof.** Let *S* be a *γ<sup>p</sup>*(*G*)-set. By Observation 1, *v* ∈ *S*. Thus *u* ∈ *S* for otherwise *u* would have two neighbors in *S*. Hence *S* is a PDS of *G* and so *γ<sup>p</sup>*(*G*) ≤ *<sup>γ</sup><sup>p</sup>*(*G*). On the other hand, by Observation 1, any *γ<sup>p</sup>*(*G*)-set contains both *u* and *v*, and thus remains a PDS of *G*. It follows that *γ<sup>p</sup>*(*G*) ≥ *<sup>γ</sup><sup>p</sup>*(*G*), and the desired equality is obtained.

Since *u* is an end strong support vertex in *G*, *u* ∈ *WR*,<sup>1</sup> *G* . By Proposition 1, there is a *γpR*(*G*)-function *f* such that *f*(*u*) = 2, and clearly *f* restricted to *G* is a PRDF on *G* yielding *γpR*(*G*) ≥ *<sup>γ</sup>pR*(*G*).

Now, assume that *u* ∈ *WR*,<sup>1</sup> *G* and let *g* be a *γpR*(*G*)-function with *g*(*u*) = 2. Then *g* can be extended to a PRDF on *G* by assigning a 0 to *x*. Thus *γpR*(*G*) ≤ *<sup>γ</sup>pR*(*G*), and the desired equality follows. 

**Proposition 5.** *Let G be a graph and u a vertex of G*. *If G is a graph obtained from G by adding a double star DS*2,2 *attached at u by one of its leaves, then:*


**Proof.** Let *x*, *y* be the non-leaf vertices of the double star *DS*2,2, and let *Lx* = {*<sup>x</sup>*1, *<sup>x</sup>*2} and *Ly* = {*y*1, *y*2}. We assume that *x*1*u* ∈ *<sup>E</sup>*(*G*).

1. For a *γ<sup>p</sup>*(*G*)-set *S*, let *S* = *S* ∪ {*<sup>x</sup>*, *y*} if *u* ∈ *S*, and *S* = *S* ∪ {*<sup>x</sup>*1, *x*, *y*} if *u* ∈ *S*. Clearly, *S* is a PDS of *G* and thus *γ<sup>p</sup>*(*G*) ≤ *γ<sup>p</sup>*(*G*) + 3.

Consider now a *γpR*(*G*)-function *f* such that *f*(*y*) = 2 (according to Proposition 1). Clearly, *f*(*x*) + *f*(*<sup>x</sup>*2) ≥ 1. If *f*(*<sup>x</sup>*1) ≤ 1, then *f* restricted to *G* is a PRDF on *G* of weight at most *γpR*(*G*) − 3 and thus *γpR*(*G*) ≥ *γpR*(*G*) + 3. If *f*(*<sup>x</sup>*1) = 2, then *f*(*u*) = 0 and the function *g* : *V*(*G*) → {0, 1, 2} defined by *g*(*u*) = 1 and *g*(*w*) = *f*(*w*) otherwise, is a PRDF on*G* of weight at most *γpR*(*G*) − 4 yielding *γpR*(*G*) ≥ *γpR*(*G*) + 4. In any case we have *γpR*(*G*) ≥ *γpR*(*G*) + 3.


**Proposition 6.** *Let G be a graph and let u be an end strong support vertex of degree 3 whose non-leaf neighbor is a support vertex, say v*, *of degree 3, where* |*Lv*| = 1. *Let G be a graph obtained from G by adding four vertices, where two are attached to a leaf of u and the other two are attached to the leaf of v. Then γ<sup>p</sup>*(*G*) = *γ<sup>p</sup>*(*G*) + 2 *and γpR*(*G*) = *γpR*(*G*) + 2*.*

**Proof.** Let *Lu* = {*<sup>x</sup>*, *x*} and *Lv* = {*y*}. Let *x*1, *x*2, *y*1 and *y*2 be the four added vertices, where *xx*1, *xx*2, *yy*1, *yy*2 ∈ *<sup>E</sup>*(*G*). By items 1 and 2 of Observation 1, any *γ<sup>p</sup>*(*G*)-set contains *u* and *v*. Clearly such a set can be extended to a PDS of *G* by adding *x*, *y* which yields *γ<sup>p</sup>*(*G*) ≤ *γ<sup>p</sup>*(*G*) + 2. On the other hand, let *D* be a *γ<sup>p</sup>*(*G*)-set. Then by items 1 and 2 of Observation 1, we have *x*, *u*, *y*, *v* ∈ *D*, and thus *D* \ {*<sup>x</sup>*, *y*} is a PDS of *G*, implying that *γ<sup>p</sup>*(*G*) ≥ *γ<sup>p</sup>*(*G*) + 2. Therefore *γ<sup>p</sup>*(*G*) = *γ<sup>p</sup>*(*G*) + 2.

Next we shall show that *γpR*(*G*) = *γpR*(*G*) + 2. First we show that *γpR*(*G*) ≤ *γpR*(*G*) + 2. Since *u* is an end strong support vertex of *G*, let *f* be a *γpR*(*G*)-function with *f*(*u*) = 2 (by Proposition 1) such that *f*(*v*) is as small as possible. If *f*(*v*) ≤ 1, then *f*(*y*) = 1, and thus the function *g* : *V*(*G*) → {0, 1, 2} defined by *g*(*x*) = *g*(*y*) = 2, *g*(*x*) = 1, *g*(*u*) = *g*(*<sup>x</sup>*1) = *g*(*<sup>x</sup>*2) = *g*(*y*1) = *g*(*y*2) = 0 and *g*(*w*) = *f*(*w*) otherwise, is a PRDF on *G*. Hence *γpR*(*G*) ≤ *γpR*(*G*) + 2. If *f*(*v*) = 2, then by our choice of *f* , we have *f*(*z*) = 0 for any *z* ∈ *<sup>N</sup>*(*v*) \ {*u*} and thus the function *h* : *V*(*G*) → {0, 1, 2} defined by *h*(*z*) = 1 for *z* ∈ *<sup>N</sup>*(*v*) \ {*<sup>u</sup>*, *y*} and *<sup>h</sup>*(*x*) = 1, *h*(*x*) = *h*(*y*) = 2, *h*(*u*) = *h*(*v*) = *h*(*<sup>x</sup>*1) = *h*(*<sup>x</sup>*2) = *h*(*y*1) = *h*(*y*2) = 0 and *h*(*w*) = *f*(*w*) otherwise, is a PRDF on *G* yielding *γpR*(*G*) ≤ *γpR*(*G*) +2. Hence *γpR*(*G*) ≤ *γpR*(*G*) +2. Now we show that *γpR*(*G*) ≥ *γpR*(*G*) + 2. By Proposition 1, let *g* be a *γpR*(*G*)-function such that *g*(*x*) = *g*(*y*) = 2. It can be seen that *g*(*x*) = 1. If *f*(*v*) = 0, then the function *h* : *V*(*G*) → {0, 1, 2} defined by *h*(*u*) = 2, *h*(*y*) = 1, *h*(*x*) = *<sup>h</sup>*(*x*) = 0 and *h*(*w*) = *g*(*w*) otherwise, is a PRDF on *G* of weight at most *γpR*(*G*) − 2. If *f*(*v*) ≥ 1, then the function *h* : *V*(*G*) → {0, 1, 2} defined by *h*(*u*) = *h*(*v*) = 2, *h*(*x*) = *<sup>h</sup>*(*x*) = *h*(*y*) = 0 and *h*(*w*) = *g*(*w*) otherwise, is a PRDF on *G* of weight at most *γpR*(*G*) − 2. In any case, *γpR*(*G*) ≥ *γpR*(*G*) + 2, and the equality follows.

### **3. The Family** T

In this section, we define the family T of unlabeled trees *T* that can be obtained from a sequence *T*1, *T*2, ... , *Tk* (*k* ≥ 1) of trees such that *T*1 ∈ {*<sup>P</sup>*2, *<sup>P</sup>*3} and *T* = *Tk*. If *k* ≥ 2, then *Ti*+<sup>1</sup> is obtained recursively from *Ti* by one of the following operations.

**Operation** O1**:** If *u* ∈ *V*(*Ti*) is an end strong support vertex, then O1 adds a vertex *x* attached at *u* by an edge *ux* to obtain *Ti*+1.

**Operation** O2**:** If *u* ∈ (*WR*,<sup>1</sup> *Ti* ∪ *WR*,≥<sup>1</sup> *Ti* ) ∩ *WAPD Ti* , then O2 adds a path *P*2 = *x*1*x*2 attached at *u* by an edge *ux*1 to obtain *Ti*+1.

**Operation** O3**:** If *u* ∈ *WR*,≤<sup>1</sup> *Ti* ∩ *WP*,*<sup>A</sup> Ti* ∩ *WAPD Ti* , then O3 adds a star *<sup>K</sup>*1,3 centered at *x* by attaching one of its leaves, say *x*1, to *u* to obtain *Ti*+1.

**Operation** O4**:** If *u* ∈ *WR*,<sup>1</sup> *Ti* is an end support vertex which is adjacent to a strong support vertex, then O4 adds a vertex *x* attached at *u* by an edge *ux* to obtain *Ti*+1.

**Operation** O5**:** If *u* ∈ *WR*,<sup>1</sup> *Ti* ∩ *WP*,*<sup>A</sup> Ti* ∩ *WAPD Ti* , then O5 adds a double star *DS*2,2 by attaching one of its leaves, say *x*1, to *u* to obtain *Ti*+1.

**Operation** O6**:** If *u* ∈ *V*(*Ti*) is an end strong support vertex of degree 3 with *x* ∈ *Lu* such that *u* is adjacent to a support vertex *v* of degree 3 with *Lv* = {*y*}, then O6 adds four vertices *x*1, *x*2, *y*1, *y*2 attached at *x* and *y* by edges *xx*1, *xx*2, *yy*1, *yy*2 to obtain *Ti*+1.

**Lemma 1.** *If Ti is a tree with γpR*(*Ti*) = *γ<sup>p</sup>*(*Ti*) + 1 *and Ti*+<sup>1</sup> *is a tree obtained from Ti by one of the Operations* O1,..., O6*, then <sup>γ</sup>pR*(*Ti*+<sup>1</sup>) = *<sup>γ</sup><sup>p</sup>*(*Ti*+<sup>1</sup>) + 1*.*

**Proof.** If *Ti*+<sup>1</sup> is obtained from *Ti* by Operation O1, then by Corollary 1 and the assumption *γpR*(*Ti*) = *γ<sup>p</sup>*(*Ti*) + 1, we have *<sup>γ</sup>pR*(*Ti*+<sup>1</sup>) = *γpR*(*Ti*) = *γ<sup>p</sup>*(*Ti*) + 1 = *<sup>γ</sup><sup>p</sup>*(*Ti*+<sup>1</sup>) + 1. If *Ti*+<sup>1</sup> is obtained from *Ti* by Operation O2, then as above the result follows from Proposition 2 (items 2, 3 and 4). If *Ti*+<sup>1</sup> is obtained from *Ti* by Operation O3, then the result follows from Proposition 3 (items 2 and 3). If *Ti*+<sup>1</sup> is obtained from *Ti* by Operation O4, then the result follows from Proposition 4. If *Ti*+<sup>1</sup> is obtained from *Ti* by Operation O5, then the result follows from Proposition 5. Finally, if *Ti*+<sup>1</sup> is obtained from *Ti* by Operation O6, then the result follows from Proposition 6.

In the rest of the paper, we shall prove our main result:

**Theorem 1.** *For any tree T of order n* ≥ 2*,*

$$
\gamma\_R^p(T) \ge \gamma^p(T) + 1\_\prime
$$

*with equality if and only if T* ∈ T *.*

### **4. Proof of Theorem 1**

**Lemma 2.** *If T* ∈ T *, then γpR*(*T*) = *γ<sup>p</sup>*(*T*) + 1*.*

**Proof.** Let *T* be a tree of T . Then there exists a sequence of trees *T*1, *T*2, ... , *Tk* (*k* ≥ 1) such that *T*1 ∈ {*<sup>P</sup>*2, *<sup>P</sup>*3}, and if *k* ≥ 2, then *Ti*+<sup>1</sup> can be obtained from *Ti* by one of the aforementioned operations. We proceed by induction on the number of operations used to construct *T*. If *k* = 1, then *T* ∈ {*<sup>P</sup>*2, *<sup>P</sup>*3} and clearly *γpR*(*T*) = *γ<sup>p</sup>*(*T*) + 1. This establishes our basis case. Let *k* ≥ 2 and assume that the result holds for each tree *T* ∈ T which can be obtained from a sequence of operations of length *k* − 1 and let *T* = *Tk*−1. By the induction hypothesis, *γpR*(*T*) = *γ<sup>p</sup>*(*T*) + 1. Since *T* = *Tk* is obtained from *T* by one of the Operations O*i* (*i* ∈ {1, 2, . . . , 6}) , we conclude from Lemma 1 that *γpR*(*T*) = *γ<sup>p</sup>*(*T*) + 1.

**Theorem 2.** *For any tree T of order n* ≥ 2*,*

$$
\gamma\_R^p(T) \ge \gamma^p(T) + 1\_\prime
$$

*with equality only if T* ∈ T *.*

**Proof.** We use an induction on *n*. If *n* ∈ {2, <sup>3</sup>}, then *T* ∈ {*<sup>P</sup>*2, *<sup>P</sup>*3}, where *γpR*(*T*) = 2 = *γ<sup>p</sup>*(*T*) + 1 and *T* ∈ T . If *n* = 4 and diam(*T*) = 2, then *T* = *<sup>K</sup>*1,3, where *γpR*(*T*) = 2 = *γ<sup>p</sup>*(*T*) + 1 and *T* ∈ T because it can be obtained from *P*3 by applying Operation O1. If *n* = 4 and diam(*T*) = 3, then *T* = *P*4, where *γpR*(*T*) = 3 = *γ<sup>p</sup>*(*T*) + 1 and clearly *T* ∈ T since it can be obtained from *P*2 by Operation O2. Let *n* ≥ 5 and assume that every tree *T* of order *n* with 2 ≤ *n* < *n* satisfies *γpR*(*T*) ≥ *γ<sup>p</sup>*(*T*) + 1 with equality only if *T* ∈ T .

Let *T* be a tree of order *n*. If diam(*T*) = 2, then *T* is a star, where *γpR*(*T*) = 2 = *γ<sup>p</sup>*(*T*) + 1 and *T* ∈ T because *T* it can be obtained from *P*3 by frequently use of Operation O1. Hence assume that diam(*T*) = 3, and thus *T* is a double star *DSp*,*q*,(*<sup>q</sup>* ≥ *p* ≥ <sup>1</sup>). If *T* = *DS*1,*q* (*q* ≥ <sup>2</sup>), then *γpR*(*T*) = 3 = *γ<sup>p</sup>*(*T*) + 1 and *T* ∈ T since it is obtained from *P*3 by applying Operation O2. If *T* = *DSp*,*q*,(*<sup>q</sup>* ≥ *p* ≥ <sup>2</sup>), then *γ<sup>p</sup>*(*T*) = 2, *γpR*(*T*) = 4 and so *γpR*(*T*) > *γ<sup>p</sup>*(*T*) + 1. Henceforth, we assume that diam(*T*) ≥ 4. Let *v*1*v*2 ... *vk* (*k* ≥ 5) be a diametrical path in *T* such that deg*T*(*<sup>v</sup>*2) is as large as possible. Root *T* at *vk* and consider the following cases.

**Case 1.** deg*T*(*<sup>v</sup>*2) ≥ 4.

Let *T* = *T* − *v*1. By Corollary 1 and the induction hypothesis on *T*, we obtain

$$
\gamma\_R^p(T) = \gamma\_R^p(T') \ge \gamma^p(T') + 1 = \gamma^p(T) + 1.
$$

Further if *γpR*(*T*) = *γ<sup>p</sup>*(*T*) + 1, then we have equality throughout this inequality chain. In particular, *γpR*(*T*) = *γ<sup>p</sup>*(*T*) + 1. By induction on *T*, we have *T* ∈ T . It follows that *T* ∈ T since it can be obtained from *T* by applying operation O1.

**Case 2.** deg*T*(*<sup>v</sup>*2) = deg*T*(*<sup>v</sup>*3) = 2.

Let *T* = *T* − *Tv*3 . For a *γ<sup>p</sup>*(*T*)-set *S*, let *S* = *S* ∪ {*<sup>v</sup>*1} if *v*4 ∈ *S* and *S* = *S* ∪ {*<sup>v</sup>*2} for otherwise. Clearly *S* is a PDS of *T* and thus *γ<sup>p</sup>*(*T*) ≤ *γ<sup>p</sup>*(*T*) + 1. Consider now a *γpR*(*T*)-function *f* . If *f*(*<sup>v</sup>*3) ∈ {0, <sup>1</sup>}, then *f*(*<sup>v</sup>*1) + *f*(*<sup>v</sup>*2) = 2 and the function *f* , restricted to *T* is a PRDF on *T* of weight at most *γpR*(*T*) − 2 . If *f*(*<sup>v</sup>*3) = 2, then *f*(*<sup>v</sup>*4) = 0 and the function *g* : *V*(*T*) → {0, 1, 2} defined by *g*(*<sup>v</sup>*4) = 1 and *g*(*z*) = *f*(*z*) otherwise, is a PRDF on *T*. In any case, *γpR*(*T*) ≥ *γpR*(*T*) + 2. By the induction hypothesis on *T*, we obtain

$$
\gamma\_R^p(T) \ge \gamma\_R^p(T') + 2 \ge \gamma^p(T') + 1 + 2 \ge \gamma^p(T) - 1 + 3 > \gamma^p(T) + 1.
$$

**Case 3.** deg*T*(*<sup>v</sup>*2) = 2 and deg*T*(*<sup>v</sup>*3) ≥ 3.

Let *T* = *T* − *Tv*2 . By Proposition 2, we have *γ<sup>p</sup>*(*T*) ≤ *γ<sup>p</sup>*(*T*) + 1 and *γpR*(*T*) ≥ *γpR*(*T*) + 1. It follows from the induction hypothesis that

$$
\gamma\_R^p(T) \ge \gamma\_R^p(T') + 1 \ge \gamma^p(T') + 1 + 1 \ge \gamma^p(T) + 1.
$$

Further if *γpR*(*T*) = *γ<sup>p</sup>*(*T*) + 1, then we have equality throughout this inequality chain. In particular, *γ<sup>p</sup>*(*T*) = *γ<sup>p</sup>*(*T*) + 1, *γpR*(*T*) = *γpR*(*T*) + 1 and *γpR*(*T*) = *γ<sup>p</sup>*(*T*) + 1. By induction on *T*, we deduce that *T* ∈ T . Next, we shall show that *v*3 ∈ (*WR*,<sup>1</sup> *T* ∪ *WR*,≥<sup>1</sup> *T* ) ∩ *WAPD T* . Let *f* be a *γpR*(*T*)-function. If *f*(*<sup>v</sup>*3) = 2, then *f*(*<sup>v</sup>*1) = 1 and *f*(*<sup>v</sup>*2) = 0 and the function *f* |*V*(*T*) is a *γpR*(*T*)-function with *f*(*<sup>v</sup>*3) = 2

and hence *v*3 ∈ *WR*,<sup>1</sup> *T* . Hence, assume that *f*(*<sup>v</sup>*3) ≤ 1. Then *f*(*<sup>v</sup>*1) + *f*(*<sup>v</sup>*2) = 2. If *f*(*<sup>v</sup>*2) ≤ 1 or *f*(*<sup>v</sup>*2) = 2 and *f*(*<sup>v</sup>*3) = 1, then the function *f* restricted to *T* is a PRDF on *T* of weight *γpR*(*T*) − 2, contradicting the fact *γpR*(*T*) = *γpR*(*T*) + 1. Hence we assume *f*(*<sup>v</sup>*2) = 2 and *f*(*<sup>v</sup>*3) = 0. Then the function *g* : *V*(*T*) → {0, 1, 2} defined by *g*(*<sup>v</sup>*3) = 1 and *g*(*x*) = *f*(*x*) otherwise, is a *γpR*(*T*)-function and so *v*3 ∈ *WR*,≥<sup>1</sup> *T* . Hence *v*3 ∈ *WR*,<sup>1</sup> *T* ∪ *WR*,≥<sup>1</sup> *T* . It remains to show that *v*3 ∈ *WAPD T* . Suppose that *v*3 ∈ *<sup>W</sup>*5*T* and let *S* be an almost PDS of *T* of size less that *<sup>γ</sup><sup>p</sup>*(*T*). Clearly, *v*3 ∈ *S* and *v*3 has no neighbor in *S*. Therefore, *S* ∪ {*<sup>v</sup>*2} is a PDS of *T* of size at most *γ<sup>p</sup>*(*T*) = *γ<sup>p</sup>*(*T*) − 1, a contradiction. Hence *v*3 ∈ *WAPD T* . It follows that *T* ∈ T since it can be obtained from *T* by Operation O2.

$$\text{Case 4. deg}\_T(v\_2) = 3.$$

Let *Lv*2 = {*<sup>v</sup>*1, *<sup>w</sup>*}. According to Cases 1, 2 and 3, we may assume that any end support vertex on a diametrical path has degree 3. Consider the following subcases.

$$\textbf{Subcase 4.1. deg}\_T(v\_3) = 2.$$

Let *T* = *T* − *Tv*3 . By Proposition 3-(1) and the induction hypothesis we have:

$$
\gamma\_R^p(T) \ge \gamma\_R^p(T') + 2 \ge \gamma^p(T') + 1 + 2 \ge \gamma^p(T) + 1.
$$

Further if *γpR*(*T*) = *γ<sup>p</sup>*(*T*) + 1, then we have equality throughout this inequality chain. In particular, *γpR*(*T*) = *γpR*(*T*) + 2, *γ<sup>p</sup>*(*T*) = *γ<sup>p</sup>*(*T*) + 2 and *γpR*(*T*) = *γ<sup>p</sup>*(*T*) + 1. It follows from the induction hypothesis that *T* ∈ T . In the next, we shall show that *v*4 ∈ *WR*,≤<sup>1</sup> *T* ∩ *WP*,*<sup>A</sup> T* ∩ *WAPD T* .

Suppose that *v*4 ∈ *WP*,*<sup>A</sup> T* and let *S* be a *γ<sup>p</sup>*(*T*)-set that does not contain *v*4. Then *S* ∪ {*<sup>v</sup>*2} is a PDS of *T*, contradicting the fact *γ<sup>p</sup>*(*T*) = *γ<sup>p</sup>*(*T*) + 2. Hence *v*4 ∈ *WP*,*<sup>A</sup> T* . Suppose now that *v*4 ∈ *WAPD T* and let *D* be an almost PDS of *T* with respect to *v*4 such that |*D*| < *<sup>γ</sup><sup>p</sup>*(*T*). Then *v*4 ∈ *D* and *v*4 has no neighbor in *D*, and thus *D* ∪ {*<sup>v</sup>*2, *<sup>v</sup>*3} is a PDS of *T* of cardinality less *γ<sup>p</sup>*(*T*) + 2, a contradiction. Hence *v*4 ∈ *WAPD T* . It remains to show that *v*4 ∈ *WR*,≤<sup>1</sup> *T* . By Proposition 1, let *f* be a *γpR*(*T*)-function such that *f*(*<sup>v</sup>*2) = 2. If *f*(*<sup>v</sup>*4) = 2, then we must have *f*(*<sup>v</sup>*3) ≥ 1. But *f* restricted to *T* is a PRDF on *T* of weight at most *γpR*(*T*) − 3, contradicting *γpR*(*T*) = *γpR*(*T*) + 2. Hence *f*(*<sup>v</sup>*4) ≤ 1. If *f*(*<sup>v</sup>*4) = 0 and *f*(*<sup>v</sup>*3) = 2, then the function *g* : *V*(*T*) → {0, 1, 2} defined by *g*(*<sup>v</sup>*4) = 1 and *g*(*x*) = *f*(*x*) otherwise, is a PRDF of *T* of weight at most *γpR*(*T*) − 3, a contradiction as above. Thus *f*(*<sup>v</sup>*4) = 1 or *f*(*<sup>v</sup>*4) = 0 and *f*(*<sup>v</sup>*3) ≤ 1. Then *f* restricted to *T* is a *γpR*(*T*)-function showing that *v*4 ∈ *WR*,≤<sup>1</sup> *T* . Hence *v*4 ∈ *WR*,≤<sup>1</sup> *T* ∩ *WP*,*<sup>A</sup> T* ∩ *WAPD T* . Therefore, *T* ∈ T because it can be obtained from *T* by Operation O3.

**Subcase 4.2.** deg*T*(*<sup>v</sup>*3) ≥ 3.

**(b)**

We distinguish between some situations.

**(a)** *v*3 is a strong support vertex.

> Let *T* = *T* − *v*1. By Proposition 4 and the induction hypothesis we have:

$$
\gamma\_R^p(T) \ge \gamma\_R^p(T') \ge \gamma^p(T') + 1 = \gamma^p(T) + 1.
$$

Further if *γpR*(*T*) = *γ<sup>p</sup>*(*T*) + 1, then we have equality throughout this inequality chain. In particular, *γpR*(*T*) = *<sup>γ</sup>pR*(*T*), *γ<sup>p</sup>*(*T*) = *γ<sup>p</sup>*(*T*) and *γpR*(*T*) = *γ<sup>p</sup>*(*T*) + 1. By the induction hypothesis, *T* ∈ T . To show *v*2 ∈ *WR*,<sup>1</sup> *T* , let *f* be a *γpR*(*T*)-function such that *f*(*<sup>v</sup>*2) = 2 (by Proposition 1). Since *γpR*(*T*) = *<sup>γ</sup>pR*(*T*), *f* is also a *γpR*(*T*)-function with *f*(*<sup>v</sup>*2) = 2, implying that *v*2 ∈ *WR*,<sup>1</sup> *T* . Therefore *T* ∈ T because it can be obtained from *T* by Operation O4. *v*3hastwochildren*x*,*y*withdepthone,differentfrom*v*2.

 Then *u* and *w* are both strong support vertices of degree 3. Let *T* = *T* − *Tv*2 . By Observation 1, any *γ<sup>p</sup>*(*T*)-set *S* contains *x* and *y* and thus *v*3 ∈ *S*. Hence *S* ∪ {*<sup>v</sup>*2} is a PDS of *T* yielding *γ<sup>p</sup>*(*T*) ≤ *γ<sup>p</sup>*(*T*) + 1. Now, let *f* be a *γpR*(*T*) function such that *f*(*<sup>v</sup>*2) = 2 and *f*(*x*) = 2

(by Proposition 1). Then *f*(*<sup>v</sup>*3) ≥ 1. It follows that the function *f* restricted to *T* is a PRDF on *T* of weight *γpR*(*T*) − 2, and hence *γpR*(*T*) ≥ *γpR*(*T*) + 2. By the induction hypothesis we have

> *γpR*(*T*) ≥ *γpR*(*T*) + 2 ≥ *γ<sup>p</sup>*(*T*) + 3 ≥ *γ<sup>p</sup>*(*T*) + 2 > *γ<sup>p</sup>*(*T*) + 1.

**(c)** *v*3 is a support vertex and has a child *u* with depth one different from *v*2.

Let *w*1 be the unique leaf adjacent to *v*3. Note that *u* is a strong support vertices of degree 3. Let *T* = *T* − *Tv*2 . If *S* is a *γ<sup>p</sup>*(*T*)-set, then by Observation 1-(2), *v*3 ∈ *S* and thus *S* ∪ {*<sup>v</sup>*2} is a PDS of *T* yielding *γ<sup>p</sup>*(*T*) ≤ *γ<sup>p</sup>*(*T*) + 1. By Proposition 1, let *f* be a *γpR*(*T*)-function such that *f*(*<sup>v</sup>*2) = 2 and *f*(*u*) = 2. By the definition of perfect Roman dominating functions, we have *f*(*<sup>v</sup>*3) ≥ 1. Then, the function *f* restricted to *T* is a PRDF on *T* of weight *γpR*(*T*) − 2 and thus *γpR*(*T*) ≥ *γpR*(*T*) + 2. It follows from the induction hypothesis that

$$
\gamma\_R^p(T) \ge \gamma\_R^p(T') + 2 \ge \gamma^p(T') + 3 \ge \gamma^p(T) + 2 > \gamma^p(T) + 1.
$$

According to (a), (b) and (c), we can assume for the next that deg*T*(*<sup>v</sup>*3) = 3. **(d)**deg*T*(*<sup>v</sup>*3)=3and*v*3hasachild*x*with depthonedifferentfrom*v*2.

Note that *x* is a strong support vertices of degree 3. Let *Lx* = {*<sup>x</sup>*1, *<sup>x</sup>*2} and let *T* be the tree obtained from *T* by removing the set of vertices {*<sup>v</sup>*1, *v*2, *w*, *x*, *x*1, *<sup>x</sup>*2}. For a *γ<sup>p</sup>*(*T*)-set *S*, let *S* = *S* ∪ {*<sup>v</sup>*2, *x*} if *v*3 ∈ *S* and *S* = *S* ∪ {*<sup>v</sup>*2, *v*3, *x*} when *v*3 ∈ *S*. Clearly, *S* is a PDS of *T* and so *γ<sup>p</sup>*(*T*) ≤ *γ<sup>p</sup>*(*T*) + 3. Now let *f* be a *γpR*(*T*)-function such that *f*(*<sup>v</sup>*2) = *f*(*x*) = 2. Then *f*(*<sup>v</sup>*3) ≥ 1 and the function *f* restricted to *T* is a PRDF on *T* of weight at most *γpR*(*T*) − 4. By the induction hypothesis we have:

$$\gamma\_R^p(T) \ge \gamma\_R^p(T') + 4 \ge \gamma^p(T') + 1 + 4 \ge \gamma^p(T) - 3 + 5 \ge \gamma^p(T) + 1.$$

**(e)** deg*T*(*<sup>v</sup>*3) = 3 and *v*3 is adjacent to exactly one leaf *<sup>w</sup>*.

If *v*4 has a child *s* with depth one and degree two, then let *T* be the tree obtained from *T* by removing *s* and its unique leaf. This case can be treated in the same way as in Case 3. Moreover, if *v*4 has a child *s* with depth one and degree at least four, then let *T* be the tree obtained from *T* by removing a leaf neighbor of *s*. This case can be treated in the same way as in Case 1. Hence, we may assume that each child of *v*4 is a leaf or a vertex with depth one and degree 3 or a vertex with depth two whose maximal subtree is isomorphic to *Tv*3 . First assume that deg*T*(*<sup>v</sup>*4) ≥ 4, and let *T* = *T* − *Tv*3 . Clearly, any *γ<sup>p</sup>*(*T*)-set contains *v*4 and such a set can be extended to a PDS of *T* by adding *v*2, *v*3. Hence *γ<sup>p</sup>*(*T*) ≤ *γ<sup>p</sup>*(*T*) + 2. Now let *f* be a *γpR*(*T*)-function such that *f*(*<sup>v</sup>*2) = 2. Clearly, *f*(*<sup>v</sup>*3) + *f*(*w*) ≥ 1. If *f*(*<sup>v</sup>*3) ≤ 1 or *f*(*<sup>v</sup>*3) = 2 and *f*(*<sup>v</sup>*4) ≥ 1, then the function *f* restricted to *T* is a PRDF on *T* and thus *γpR*(*T*) ≥ *γpR*(*T*) + 3. Hence assume that *f*(*<sup>v</sup>*3) = 2 and *f*(*<sup>v</sup>*4) = 0. Then the function *g* : *V*(*T*) → {0, 1, 2} defined by *g*(*<sup>v</sup>*4) = 1 and *g*(*u*) = *f*(*u*) otherwise, is a PRDF of *T* of weight *γpR*(*T*) − 3 and thus *γpR*(*T*) ≥ *γpR*(*T*) + 3. By the induction hypothesis we have:

$$\gamma\_R^p(T) \ge \gamma\_R^p(T') + 3 \ge \gamma^p(T') + 1 + 3 \ge \gamma^p(T) - 2 + 4 > \gamma^p(T) + 1.$$

From now on, we can assume that deg*T*(*<sup>v</sup>*4) ≤ 3. We examine different cases.

(e.1.) *v*4 has a child *x* of degree 3 and depth 1.

> Let *Lx* = {*<sup>z</sup>*1, *<sup>z</sup>*2} and let *T* be the tree obtained from *T* by removing the set {*<sup>v</sup>*1, *w*, *z*1, *<sup>z</sup>*2}. By Proposition 6, we have *γ<sup>p</sup>*(*T*) = *γ<sup>p</sup>*(*T*) + 2 and *γpR*(*T*) = *γpR*(*T*) + 2. We deduce from the induction hypothesis that

$$
\gamma\_R^p(T) = \gamma\_R^p(T') + 2 \ge \gamma^p(T') + 1 + 2 = \gamma^p(T) - 2 + 3 = \gamma^p(T) + 1.
$$

Further if *γpR*(*T*) = *γ<sup>p</sup>*(*T*) + 1, then we have equality throughout this inequality chain. In particular, *γpR*(*T*) = *γ<sup>p</sup>*(*T*) + 1. By induction on *T*, we have *T* ∈ T . Therefore *T* ∈ T since it can be obtained from *T* by Operation O6.

### (e.2.) *v*4 has a child *v*3with depth two.

Note that *Tv*3 and *Tv*3 are isomorphic. Let *T* = *T* − (*Tv*3 ∪ *Tv*3 ), and observe that *v*4 is a leaf in *T*. Since any *γ<sup>p</sup>*(*T*)-set can be extended to a PDS of *T* by adding *v*4 and the support vertices of *Tv*3 ∪ *Tv*3 we obtain *γ<sup>p</sup>*(*T*) ≤ *γ<sup>p</sup>*(*T*) + 5. Moreover, as above we can see that *γpR*(*T*) ≥ *γpR*(*T*) + 6. Now, by induction hypothesis we obtain:

$$
\gamma\_R^p(T) \ge \gamma\_R^p(T') + 6 \ge \gamma^p(T') + 1 + 6 \ge \gamma^p(T) - 5 + 7 > \gamma^p(T) + 1.
$$

(e.3.) deg*T*(*<sup>v</sup>*4) = 2.

> Let *T* = *T* − *Tv*4 . If *V*(*T*) = {*<sup>v</sup>*5}, then it can be seen that *T* is tree with *γpR*(*T*) = 5 and *γ<sup>p</sup>*(*T*) = 3, implying that *γpR*(*T*) > *γ<sup>p</sup>*(*T*) + 1. Hence we assume that *T* is nontrivial. By Proposition 5 and by the inductive hypothesis we have:

$$
\gamma\_\mathcal{R}^p(T) \ge \gamma\_\mathcal{R}^p(T') + 3 \ge \gamma^p(T') + 1 + 3 \ge \gamma^p(T) - 3 + 4 = \gamma^p(T) + 1.
$$

Further if *γpR*(*T*) = *γ<sup>p</sup>*(*T*) + 1, then we have equality throughout this inequality chain. In particular, *γpR*(*T*) = *γpR*(*T*) + 3, *γ<sup>p</sup>*(*T*) = *γ<sup>p</sup>*(*T*) + 3 and *γpR*(*T*) = *γ<sup>p</sup>*(*T*) + 1. By induction on *T*, we have *T* ∈ T . Next, we shall show that *v*5 ∈ *WR*,<sup>1</sup> *T* ∩ *WP*,*<sup>A</sup> T* ∩ *WAPD T* . Suppose that *v*5 ∈ *WP*,*<sup>A</sup> T* and let *S* be a *γ<sup>p</sup>*(*T*)-set that does not contain *v*5. Then *S* ∪ {*<sup>v</sup>*2, *<sup>v</sup>*3} is a PDS of *T* contradicting *γ<sup>p</sup>*(*T*) = *γ<sup>p</sup>*(*T*) + 3. Hence *v*5 ∈ *WP*,*<sup>A</sup> T* . Suppose that *v*5 ∈ *WAPD T* and let *S* be an almost PDS of *T* such that |*S*| ≤ *γ<sup>p</sup>*(*T*) − 1. Clearly, *v*5 ∈ *S* and *v*5 has no neighbor in *S*. It follows that *S* ∪ {*<sup>v</sup>*4, *v*3, *<sup>v</sup>*2} is a PDS of *T* of size |*S*| + 3 ≤ *γ<sup>p</sup>*(*T*) − 1, a contradiction. Thus *v*5 ∈ *WAPD T* . Next we show that *v*5 ∈ *WR*,<sup>1</sup> *T* . Let *f* be a *γpR*(*T*)-function such that *f*(*<sup>v</sup>*2) = 2. To Roman dominate *<sup>w</sup>*, we must have either *f*(*w*) = 1 or *f*(*<sup>v</sup>*3) = 2. We claim that *f*(*<sup>v</sup>*4) ≤ 1. Suppose, to the contrary, that *f*(*<sup>v</sup>*4) = 2. By definition of perfect Roman dominating functions, we may assume that *f*(*<sup>v</sup>*3) = 2. But then the function *g* : *V*(*T*) → {0, 1, 2} defined by *g*(*<sup>v</sup>*5) = 1 and *g*(*x*) = *f*(*x*) otherwise, is a PRDF of *T* of weight *γpR*(*T*) − 5 contradicting *γpR*(*T*) = *γpR*(*T*) + 3. Hence *f*(*<sup>v</sup>*4) ≤ 1. It follows that the function *f* restricted to *T* is a PRDF of *T* of weight at most *γpR*(*T*) − 3 for which we conclude from *γpR*(*T*) = *γpR*(*T*) + 3 that *f*(*<sup>v</sup>*3) = *f*(*<sup>v</sup>*4) = 0 and *f*(*w*) = 1. Hence to Roman dominate *v*4, we must have *f*(*<sup>v</sup>*5) = 2 and thus function *f* restricted to *T* is a *γpR*(*T*)-function that assigns a a 2 to *v*5. Hence *v*5 ∈ *WR*,<sup>1</sup> *T* , and thus *v*5 ∈ *WR*,<sup>1</sup> *T* ∩ *WP*,*<sup>A</sup> T* ∩ *WAPD T* . Therefore, *T* ∈ T since it can be obtained from *T* by Operation O5.

(e.4.) deg*T*(*<sup>v</sup>*4) = 3 and *v*4 has a child *z* with depth 0.

> Seeing the above Cases and Subcases as we did in the beginning of Case (e), we may assume that any child of *v*5 is a leaf, or an end strong support vertex of degree 3, or a vertex with depth 2 whose maximal subtree is isomorphic to *Tv*3 , or a vertex with depth 3 whose maximal subtree is isomorphic to *Tv*4 . Assume first that deg*T*(*<sup>v</sup>*5) ≥ 4, and let *T* = *T* − *Tv*4 . Clearly, *v*5 belongs to any *γ<sup>p</sup>*(*T*)-set and such a set *γ<sup>p</sup>*(*T*)-set can be extended to a PDS of *T* by adding *v*2, *v*3, *v*4, implying that *γ<sup>p</sup>*(*T*) ≤ *γ<sup>p</sup>*(*T*) + 3. Next we show that *γpR*(*T*) ≥ *γpR*(*T*) + 4. Let *f* be a *γpR*(*T*)-function such that *f*(*<sup>v</sup>*2) = 2. Clearly *f*(*<sup>v</sup>*3) + *f*(*w*) ≥ 1 and *f*(*<sup>v</sup>*4) + *f*(*z*) ≥ 1. If *f*(*<sup>v</sup>*4) ≤ 1 or *f*(*<sup>v</sup>*5) ≥ 1, then the function *f* restricted to *T* is a PRDF on *T* yielding *γpR*(*T*) ≥ *γpR*(*T*) + 4. Hence assume that *f*(*<sup>v</sup>*4) = 2 and *f*(*<sup>v</sup>*5) = 0. Then the function

*g* : *V*(*T*) → {0, 1, 2} defined by *g*(*<sup>v</sup>*5) = 1 and *g*(*u*) = *f*(*u*) otherwise, is a PRDF of *T* yielding *γpR*(*T*) ≥ *γpR*(*T*) + 4. By induction on *T*, it follows that

$$
\gamma\_R^p(T) \ge \gamma\_R^p(T') + 4 \ge \gamma^p(T') + 5 \ge \gamma^p(T) + 2 > \gamma^p(T) + 1.
$$

For the next, we assume that deg*T*(*<sup>v</sup>*5) ≤ 3. If deg*T*(*<sup>v</sup>*5) = 1, then it can be seen that *T* is a tree with *γpR*(*T*) = 6, *γ<sup>p</sup>*(*T*) and so *γpR*(*T*) > *γ<sup>p</sup>*(*T*) + 1. Hence we assume that deg*T*(*<sup>v</sup>*5) ∈ {2, <sup>3</sup>}. Consider the following situations.


### (e.4.4.) deg*T*(*<sup>v</sup>*5) = 3 and *v*5 has a children *y* with depth 1 and degree 3.

Let *T* = *T* − (*Tv*4 ∪ *Ty*). Clearly, any *γ<sup>p</sup>*(*T*)-set can be extended to a PDS of *T* by adding *v*5, *v*2, *v*3, *v*4, *y* and thus *γ<sup>p</sup>*(*T*) ≤ *γ<sup>p</sup>*(*T*) + 5. Next, we show that *γpR*(*T*) ≥ *γpR*(*T*) + 6. Let *f* be a *γpR*(*T*)-function such that *f*(*<sup>v</sup>*2) = 2 and *f*(*y*) = 2 (by Proposition 1). Clearly *f*(*<sup>v</sup>*3) + *f*(*w*) ≥ 1 and *f*(*<sup>v</sup>*4) + *f*(*z*) ≥ 1. If *f*(*<sup>v</sup>*5) ≥ 1, then the function *f* restricted to *T* is a PRDF on *T* yielding *γpR*(*T*) ≥ *γpR*(*T*) + 6. Thus, let *f*(*<sup>v</sup>*5) = 0. Then to Roman dominate *z*, *v*4, *<sup>w</sup>*, we must have *f*(*z*) + *f*(*<sup>v</sup>*4) + *f*(*<sup>v</sup>*3) + *f*(*w*) ≥ 4. Then the function *g* : *V*(*T*) → {0, 1, 2} defined by *g*(*<sup>v</sup>*5) = 1 and *g*(*u*) = *f*(*u*) otherwise, is a PRDF on *T* yielding *γpR*(*T*) ≥ *γpR*(*T*) + 6. It follows from the induction hypothesis that

$$
\gamma\_R^p(T) \ge \gamma\_R^p(T') + 6 \ge \gamma^p(T') + 7 \ge \gamma^p(T) + 2 > \gamma^p(T) + 1.
$$


Assume now that deg*T*(*<sup>v</sup>*6) ≥ 3. By above Cases and Subcases, we may assume that any child of *v*6 is a leaf, or a vertex with depth *j* whose maximal subtree is isomorphic to *Tvj*+<sup>1</sup> for *j* = 2, 3, 4. Let *T* be a tree obtained from *T* by removing *v*3, *<sup>w</sup>*, *v*4, *z*, *v*5, *z* and

joining *v*2 to *v*6. Clearly, any *γ<sup>p</sup>*(*T*)-set contains *v*2, *v*6 and such a set can be extended to a PDS of *T* by *v*3, *v*4, *v*5 yielding *γ<sup>p</sup>*(*T*) ≤ *γ<sup>p</sup>*(*T*) + 3. Now, let *f* be a *γpR*(*T*)-function, and let *r* = *f*(*<sup>v</sup>*3) + *f*(*w*) + *f*(*<sup>v</sup>*4) + *f*(*z*) + *f*(*<sup>v</sup>*5) + *f*(*z*). To Roman dominate the vertices *v*3, *<sup>w</sup>*, *v*4, *z*, *v*5, *<sup>z</sup>*, we must have *r* ≥ 5 when *f*(*<sup>v</sup>*5) ≤ 1 or *r* = 4 when *f*(*<sup>v</sup>*5) = 2. If *r* = 4 or *r* ≥ 5 and *f*(*<sup>v</sup>*6) ≥ 1, then the function *f* restricted to *T* is a PRDF on *T* implying that *γpR*(*T*) ≥ *γpR*(*T*) + 4. Hence assume that *r* ≥ 5 and *f*(*<sup>v</sup>*6) = 0. Then the function *h* : *V*(*T*) → {0, 1, 2} defined by *h*(*<sup>v</sup>*6) = 1 and *h*(*x*) = *f*(*x*) otherwise, is a PRDF on *T* yielding *γpR*(*T*) ≥ *γpR*(*T*) + 4. By the induction hypothesis we obtain

$$
\gamma\_R^p(T) \ge \gamma\_R^p(T') + 4 \ge \gamma^p(T') + 1 + 4 \ge \gamma^p(T) - 3 + 5 > \gamma^p(T) + 1,
$$

and the proof is complete.

According to Lemma 2 and Theorem 2, we have proven Theorem 1.

**Author Contributions:** Z.S. and S.M.S. contribute for supervision, methodology, validation, project administration and formal analyzing. S.K., M.C., M.S. contribute for investigation, resources, some computations and wrote the initial draft of the paper which were investigated and approved by Z.S. and M.C. wrote the final draft. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the National Key R & D Program of China (Grant No. 2019YFA0706402) and the Natural Science Foundation of Guangdong Province under gran<sup>t</sup> 2018A0303130115.

**Conflicts of Interest:** The authors declare no conflict of interest.
