*Article* **On the Double Roman Domination in Generalized Petersen Graphs** *P***(5***k***,** *k***)**

**Darja Rupnik Poklukar <sup>1</sup> and Janez Žerovnik 1,2,\***


**Abstract:** A double Roman dominating function on a graph *G* = (*V*, *E*) is a function *f* : *V* → {0, 1, 2, 3} satisfying the condition that every vertex *u* for which *f*(*u*) = 0 is adjacent to at least one vertex assigned 3 or at least two vertices assigned 2, and every vertex *u* with *f*(*u*) = 1 is adjacent to at least one vertex assigned 2 or 3. The weight of *<sup>f</sup>* equals *<sup>w</sup>*(*f*) = <sup>∑</sup>*v*∈*<sup>V</sup> <sup>f</sup>*(*v*). The double Roman domination number *γdR*(*G*) of a graph *G* equals the minimum weight of a double Roman dominating function of *G*. We obtain closed expressions for the double Roman domination number of generalized Petersen graphs *P*(5*k*, *k*). It is proven that *γdR*(*P*(5*k*, *k*)) = 8*k* for *k* ≡ 2, 3 mod 5 and 8*k* ≤ *γdR*(*P*(5*k*, *k*)) ≤ 8*k* + 2 for *k* ≡ 0, 1, 4 mod 5. We also improve the upper bounds for generalized Petersen graphs *P*(20*k*, *k*).

**Keywords:** double Roman domination; generalized Petersen graph; discharging method; graph cover; double Roman graph

#### **1. Introduction**

Double Roman domination of graphs was first studied in [1], motivated by a number of applications of Roman domination in present time and in history [2]. The initial studies of Roman domination [3,4] have been motivated by a historical application. In the 4th century, Emperor Constantine was faced with a difficult problem of how to defend the Roman Empire with limited resources. His decision was to allocate two types of armies to the provinces in such a way that all the provinces in the empire will be safe. Some military units were well trained and capable of moving rapidly from one city to another in order to respond to any attack. Other legions consisted of a local militia and they were permanently positioned in a given province. The Emperor decreed that no legion could ever leave a province to defend another if in this case they left the province undefended. Thus, at some provinces two units were stationed, a local militia units were stationed at others, and some provinces had no army. While the problem is still of interest in military operations research [5], it also has applications in cases where a time-critical service is to be provided with some backup. For example, a fire station should never send all emergency vehicles to answer a call.

Similar reasoning applies in any emergency service. Hence positioning the fire stations, first aid stations, etc. at optimal positions improves the public services without increasing the cost. A natural generalization, in particular in the case of emergency services, is the *k*-Roman domination [6], where in the district not one, but *k* emergency teams are expected to be quickly available in case of multiple emergency calls. Special case *k* = 2, the double Roman domination, is considered in this work. It is well-known that the decision version of the double Roman domination problem (MIN-DOUBLE-RDF) is NP-complete, even when restricted to planar graphs, chordal graphs, bipartite graphs, undirected path graphs, chordal bipartite graphs and to circle graphs [7–9]. It is therefore of interest to study the complexity of the problem for other families of graphs. For example, linear time

**Citation:** Rupnik Poklukar, D.; Žerovnik, J. On the Double Roman Domination in Generalized Petersen Graphs *P*(5*k*, *k*). *Mathematics* **2022**, *10*, 119. https://doi.org/10.3390/ math10010119

Academic Editor: Mikhail Goubko

Received: 25 November 2021 Accepted: 24 December 2021 Published: 1 January 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

algorithms exist for interval graphs and block graphs [8], for trees [10], for proper interval graphs [11] and for unicyclic graphs [9]. Another avenue of research that is motivated by high complexity of the problem is to obtain closed expressions for the double Roman domination number of some families of graphs. In particular, generalized Petersen graphs and certain subfamilies of generalized Petersen graphs have been studied extensively in recent years. The results listed in subsection on related previous work include closed expressions for the double Roman domination number of some, and tight bounds for other subfamilies [12–15]. For more results on double Roman domination we refer to recent papers [16–19] and the references there.

The rest of the paper is organized as follows. In Section 2, we recall some basic definitions and known results that will be used in the following sections. In the last part of Section 2 our main results, Theorems 6 and 7, are presented. In Section 3, we present upper and lower bounds for double Roman domination number in generalized Petersen graphs *P*(5*k*, *k*). Finally, in Section 4, we give an improved upper bounds for double Roman domination number of generalized Petersen graphs *P*(20*k*, *k*), using the notion of covering graphs.

#### **2. Preliminaries**

#### *2.1. Graphs and Double Roman Domination*

Let *G* = (*V*, *E*) be a graph without loops and multiple edges. As usual, we denote with *V* = *V*(*G*) the vertex set of *G* and with *E* = *E*(*G*) its edge set.

A set *D* ⊆ *V*(*G*) is a dominating set if every vertex in *V*(*G*) \ *D* has at least one neighbor in *D*. The domination number *γ*(*G*) is the cardinality of a minimum dominating set of *G*. A double Roman dominating function (DRDF) on a graph *G* = (*V*, *E*) is a function *f* : *V* → {0, 1, 2, 3} with the following properties:


Define *<sup>f</sup>*(*U*) = <sup>∑</sup>*u*∈*<sup>U</sup> <sup>f</sup>*(*u*) as the weight of *<sup>f</sup>* on an arbitrary subset *<sup>U</sup>* ⊆ *<sup>V</sup>*(*G*). Then, the weight of *<sup>f</sup>* equals *<sup>w</sup>*(*f*) = *<sup>f</sup>*(*V*(*G*)) = <sup>∑</sup>*v*∈*V*(*G*) *<sup>f</sup>*(*v*). The double Roman domination number *γdR*(*G*) of a graph *G* is the minimum weight of a double Roman dominating function of *G*. A DRD function *f* is called a *γdR*-function of *G* if *w*(*f*) = *γdR*(*G*).

For any double Roman dominating function *f* , defined on *G* we define a partition of the vertex set *<sup>V</sup>* <sup>=</sup> *<sup>V</sup>*<sup>0</sup> <sup>∪</sup> *<sup>V</sup>*<sup>1</sup> <sup>∪</sup> *<sup>V</sup>*<sup>2</sup> <sup>∪</sup> *<sup>V</sup>*3, where *Vi* <sup>=</sup> *<sup>V</sup> <sup>f</sup> <sup>i</sup>* = {*u* | *f*(*u*) = *i*}.

The study of the double Roman domination in graphs was initiated by Beeler et al. [1]. It was proved that 2*γ*(*G*) ≤ *γdR*(*G*) ≤ 3*γ*(*G*). Furthermore, Beeler at al. defined a graph *G* to be double Roman if *γdR*(*G*) = 3*γ*(*G*), where *γ*(*G*) is the domination number of *G*. For a later reference we recall the following result, also obtained by Beeler et al.

**Proposition 1** ([1])**.** *In a double Roman dominating function f of weight γdR*(*G*)*, no vertex needs to be assigned the value 1.*

Domination in graphs with its many varieties has been extensively studied in the past [20–23]. Roman domination and double Roman domination is a rather new variety of interest [1,2,7,24–27].

#### *2.2. Generalized Petersen Graphs*

The generalized Petersen graph *P*(*n*, *k*) is a graph with vertex set *U* ∪ *V* and edge set *E*<sup>1</sup> ∪ *E*<sup>2</sup> ∪ *E*3, where *U* = {*u*0, *u*1, ··· , *un*−1}, *V* = {*v*0, *v*1, ··· , *vn*−1}, *E*<sup>1</sup> = {*uiui*+<sup>1</sup> | *i* = 0, 1, ... , *n* − 1}, *E*<sup>2</sup> = {*uivi* | *i* = 0, 1, ... , *n* − 1}, *E*<sup>3</sup> = {*vivi*<sup>+</sup>*<sup>k</sup>* | *i* = 0, 1, ... , *n* − 1}, and subscripts are reduced modulo *n*, see Figure 1. Thus, we identify integers *i* and *j* iff *i* ≡ *j* mod *n*. (As usual, *m* ≡ *r* mod *n* means that *m* = *kn* + *r*, or equivalently, *m* − *r* = *kn* for some integer *k* ∈ Z.

**Figure 1.** A generalized Petersen graph *P*(*n*, *k*).

It is well known that the graphs *P*(*n*, *k*) are 3-regular unless *k* = *<sup>n</sup>* <sup>2</sup> and that *P*(*n*, *k*) are highly symmetric [28,29]. As *P*(*n*, *k*) and *P*(*n*, *n* − *k*) are isomorphic, it is natural to restrict attention to *<sup>P</sup>*(*n*, *<sup>k</sup>*) with *<sup>n</sup>* <sup>≥</sup> 3 and *<sup>k</sup>*, 1 <sup>≤</sup> *<sup>k</sup>* <sup>&</sup>lt; *<sup>n</sup>* 2 .

Petersen graphs are among the most interesting examples when considering nontrivial graph invariants. The domination and its variations (such as vertex domination, exact domination, rainbow domination, double Roman domination and other) of generalized Petersen graphs have been extensively studied in recent years, see for example [14,30–36].

#### *2.3. Related Previous Work*

The domination number for the generalized Petersen graphs *P*(*ck*, *k*) for integer constants *c* ≥ 3 was studied by Zhao et al. [37]. They obtained upper bound on *γ*(*P*(*ck*, *k*)) for general *c*.

**Theorem 1** ([37])**.** *For any k* ≥ 1 *and c* ≥ 3

$$\gamma(P(ck,k)) \le \begin{cases} \frac{c}{3} \left\lceil \frac{5k}{3} \right\rceil, & c \equiv 0 \bmod 3, \\\frac{c}{3} \left\lceil \frac{5k}{3} \right\rceil - \left\lceil \frac{2k}{3} \right\rceil, & c \equiv 1 \bmod 3, \\\frac{c}{3} \left\lceil \frac{5k}{3} \right\rceil - \left\lceil \frac{2k}{3} \right\rceil + \left\lceil \frac{k}{3} \right\rceil, & c \equiv 2 \bmod 3. \end{cases}$$

Shao et al. [14] determine the exact value of *γdR*(*P*(*n*, 1)), and Jiang et al. [13] determine *γdR*(*P*(*n*, 2)).

**Theorem 2** ([13,14])**.** *Let n* ≥ 3*. Then we have*

$$\gamma\_{dR}(P(n,1)) = \begin{cases} \frac{3n}{2}, & n \equiv 0 \bmod 4, \\\frac{3n+3}{2}, & n \equiv 1,3 \bmod 4, \\\frac{3n+4}{2}, & n \equiv 2 \bmod 4 \end{cases}$$

*and for n* ≥ 5

$$\gamma\_{dR}(P(n,2)) = \begin{cases} \lceil \frac{8n}{5} \rceil, & n \equiv 0 \bmod 5, \\\lceil \frac{8n}{5} \rceil + 1, & n \equiv 1,2,3,4 \bmod 5. \end{cases}$$

Shao et al. in [14] obtained also a general lower bound on double Roman domination numbers for arbitrary graphs of a maximum degree greater or equal one.

**Theorem 3** ([14])**.** *If G is a graph of maximum degree* ≥ 1*, then*

$$
\gamma\_{dR}(G) \ge \left\lceil \frac{3V(G)}{\triangle + 1} \right\rceil.
$$

Clearly, as the generalized Petersen graph *P*(*n*, *k*) is 3-regular and has 2*n* vertices.

**Corollary 1** ([14])**.** *In Petersen graphs P*(*n*, *<sup>k</sup>*)*, <sup>γ</sup>dR*(*P*(*n*, *<sup>k</sup>*)) ≥ <sup>3</sup>*<sup>n</sup>* <sup>2</sup> .

Gao et al. [12] determined the exact value of *γdR*(*P*(*n*, *k*)) for *n* ≡ 0 mod 4 and *k* ≡ 1 mod 2, and presented an improved upper bound for *γdR*(*P*(*n*, *k*)) in other cases. The results are summarized in the next theorem.

**Theorem 4** ([12])**.** *For k* <sup>≥</sup> <sup>3</sup>*, <sup>γ</sup>dR*(*P*(*n*, *<sup>k</sup>*)) = <sup>3</sup>*<sup>n</sup>* <sup>2</sup> *, k* ≡ 1 mod 2, *n* ≡ 0 mod 4.

$$\left[\frac{3n}{2}\right] \leq \gamma\_{dR}(\mathbb{P}(n,k)) \leq \begin{cases} \frac{3n}{2} + \frac{5k+5}{4}, & k \equiv 1 \bmod 4, n \neq 0 \bmod 4, \\\frac{3n}{2} + \frac{5k+7}{4}, & k \equiv 3 \bmod 4, n \neq 0 \bmod 4, \\\frac{3n}{2} \frac{(3k+2)}{(3k+1)'} & k \equiv 0 \bmod 4, n \equiv 0 \bmod (3k+1), \\\frac{\left[\frac{3n}{2}\left(\frac{3k+2}{(3k+1)}\right) + \frac{5k+4}{4}\right]}{2\left(\frac{3k}{2}-1\right)'} & k \equiv 0 \bmod 4, n \neq 0 \bmod (3k+1), \\\frac{3n}{2} \left(\frac{3k}{(3k-1)}\right) & k \equiv 2 \bmod 4, n \equiv 0 \bmod (3k-1), \\\left[\frac{3n}{2}\left(\frac{3k}{2}\right) + \frac{5k+6}{4}\right] & k \equiv 2 \bmod 4, n \neq 0 \bmod (3k-1). \end{cases}$$

Double Roman domination of families *P*(*ck*, *k*) has been studied recently for small *k*, including *c* = 3, 4, and 5. Shao et al. [15] considered the double Roman domination number in generalized Petersen graphs *P*(3*k*, *k*) and constructed solutions providing the upper bounds, which gives exact values for *γdR*(*P*(3*k*, *k*)).

**Theorem 5** ([15])**.**

$$\gamma\_{dR}(P(\mathfrak{B};k)) = \begin{cases} \mathfrak{S}k + 1, & k \in \{1, 2, 4\} \\ \mathfrak{S}k, & otherwise \end{cases}$$

For small cases in the families *P*(4*k*, *k*) and *P*(5*k*, *k*), the known facts are summarized in the next proposition.

**Proposition 2.** *γdR*(*P*(4, 1)) = 6*, γdR*(*P*(8, 2)) = 14 *[13], γdR*(*P*(12, 3)) = 18 *and γdR*(*P*(5, 1)) = 9 *[14], γdR*(*P*(10, 2)) = 16 *[13],* 23 ≤ *γdR*(*P*(15, 3)) ≤ 26 *[12].*

Wang et al. in [38] showed that *<sup>γ</sup>*(*P*(4*k*, *<sup>k</sup>*)) = \* <sup>2</sup>*k*; *<sup>k</sup>* <sup>≡</sup> 1 mod 2, <sup>2</sup>*<sup>k</sup>* <sup>+</sup> 1; *<sup>k</sup>* <sup>≡</sup> 0 mod 2 , and *γ*(*P*(5*k*, *k*)) = 3*k* for all *k* ≥ 1. Furthermore, recall the lower bound given in Corollary 1 and recall that Theorem 4 implies *γdR*(*P*(4*k*, *k*)) = 6*k* for *k* ≡ 1 mod 2. Thus, we can write the known facts regarding *γdR*(*P*(4*k*, *k*)) and *γdR*(*P*(5*k*, *k*)) in the next two propositions.

**Proposition 3.** *Let k* ≥ 1*. If k* ≡ 1 mod 2*, then γdR*(*P*(4*k*, *k*)) = 6*k, and if k* ≡ 0 mod 2 *then* 6*k* ≤ *γdR*(*P*(4*k*, *k*)) ≤ 6*k* + 3.

**Proposition 4.** *Let k* ≥ 3*. Then* 7*k* + & *k* 2 ' ≤ *γdR*(*P*(5*k*, *k*)) ≤ 9*k*.

#### *2.4. Our Results*

The main result of our paper are either exact values or narrow bounds for the double Roman domination numbers of all Petersen graphs *P*(5*k*, *k*). More precisely, we will show that the following theorem holds.

**Theorem 6.** *Let k* ≥ 2*.*

$$8k \le \gamma\_{dR}(P(5k,k)) \le \begin{cases} 8k, & k \equiv 2,3 \bmod 5 \\\ 8k+2, & otherwise \end{cases}$$

As mentioned earlier, a graph G is double Roman if *γdR*(*G*) = 3*γ*(*G*). Using the known equality *γ*(*P*(5*k*, *k*)) = 3*k* for all *k* ≥ 1 [38], we can conclude that the only double Roman graph in the set of generalized Petersen graphs *P*(5*k*, *k*) is *P*(5, 1).

**Corollary 2.** *There is no double Roman graphs in the set of generalized Petersen graphs P*(5*k*, *k*) *for k* ≥ 2*. The graph P*(5, 1) *is a double Roman graph.*

We also show that certain generalized Petersen graphs are covering graphs of other generalized Petersen graphs (Proposition 7). This provides a method for establishing new upper bounds, see Proposition 8. In particular, we elaborate the case *P*(20*k*, *k*) to obtain exact values in some, and tight bounds in other cases.

**Theorem 7.** *γdR*(*P*(20, 1)) ≤ 40*, γdR*(*P*(40, 2)) ≤ 64*, γdR*(*P*(60, 3)) ≤ 96*. Furthermore, let k* > 3*. Then, for odd k, we have*

$$
\gamma\_{dR}(P(\text{20k}/k)) = \text{\textquotedbl{}0k} \tag{1}
$$

*and for k even,*

$$\Im \Im 0k \le \gamma\_{dR} (P(\Im 0k, k)) \le \Im 0k + 15. \tag{2}$$

By Proposition 1, we can only consider the DRDF of a graph *G* with no vertex assigned the value 1.

#### **3. Constructions and Proofs**

In this section, the constructions of double Roman dominating functions providing upper bounds for the double Roman dominating numbers are given. We start by introducing some convenient notation for representing the DRDFs and providing some basic constructions.

In order to present the double Roman dominating functions of generalized Petersen graphs as concise as possible we use two different notations. For smaller graphs, we use the notation in brackets, showing weights on outer and inner cycles in two lines:

$$
\begin{pmatrix} f(u\_0) & f(u\_1) & \dots & f(u\_{n-1}) \\ f(v\_0) & f(v\_1) & \dots & f(v\_{n-1}) \end{pmatrix} \dots
$$

For example, a DRD function showing *γdR*(*P*(4, 1)) = 6 is the following:

 0030 3000 .

For bigger graphs we use the notation that provides only the values on the outer cycle, see Table 1. (In this case, the assignment on the inner cycles is completed such that the weight is minimal, see Lemma 1 for details.)

The columns correspond to the sets *Ui* = {*ui*, *ui*<sup>+</sup>*k*, *ui*<sup>+</sup>2*k*, ... , *ui*+(*c*−1)*k*}, and we assume that the inner cycles, sets *Vi* = {*vi*, *vi*<sup>+</sup>*k*, *vi*<sup>+</sup>2*k*,..., *vi*+(*c*−1)*k*}, are completed such that

the whole assignment presents a DRD function. As we can see from Table 1 below, the first two and the last two columns provide the same information on DRD function, namely the values at *U*<sup>0</sup> = *Uk*, and *U*<sup>1</sup> = *Uk*+1. We will use this property observing certain patterns appearing in case of optimal assignment —it will hold exactly when columns 0 and *k* will match, taking into account the shift of rows as indicated in Table 1.

**Table 1.** A DRD function of *Ui* for *P*(4*k*, *k*).


To better understand the notation in tables, consider the pattern in Table 2 that provides DRDF(double Roman dominating function) for *P*(12, 3) and *P*(28, 7).

**Table 2.** An optimal DRDF(double Roman dominating function) of *Ui* for *P*(4*k*, *k*). The first column provides a DRD function for *P*(4, 1), the first 3 columns provide a DRD function for *P*(12, 3), and the first 7 columns provide a DRD function for *P*(28, 7).


Considering closely the graph *P*(12, 3) and using the fact that the columns 0 and 4 correspond to the same set of vertices, *U*<sup>0</sup> = *U*4, and that the column 4 equals column 0 shifted one row downwards (see Table 1), we can see that the pattern is well defined on the outer cycle of *P*(12, 3). Obviously, the vertices on the inner cycles could be assigned with three more weights of 3, so we have a DRDF of *P*(12, 3) of weight 9 + 9 = 18. Similarly, regarding *P*(28, 7), we have *U*<sup>0</sup> = *U*7, and the same reasoning applies. Recalling Theorem 4, the constructions are best possible (compare the bounds in Theorem 3.)

#### *3.1. Basic Constructions for P*(5*k*, *k*)*.*

Recall that *γdR*(*P*(5, 1)) = 9 [14]. A simple DRD function showing *γdR*(*P*(5, 1)) ≤ 9 is the following:

$$
\begin{pmatrix} 3 & 0 & 0 & 0 & 3 \\ 0 & 0 & 3 & 0 & 0 \end{pmatrix} \dots
$$

For larger graphs among *P*(5*k*, *k*), we are going to use the notation introduced in Table 3.

**Table 3.** A DRD function of *Ui* for *P*(5*k*, *k*).


The next two examples provide constructions of DRDF that show *γdR*(*P*(10, 2)) ≤ 16 and *γdR*(*P*(35, 7)) ≤ 56.

We proceed with two comments on Table 4.



**Table 4.** A DRDF of *Ui* that implies *γdR*(*P*(10, 2)) ≤ 16 and *γdR*(*P*(35, 7)) ≤ 56.

These constructions are optimal, which will follow from the lower bound that will be proved below. Recall that *γdR*(*P*(10, 2)) = 16 [13], therefore the RDF for *γdR*(*P*(10, 2)) given in Table 4 is best possible.

A symmetrical construction, given in Table 5 shows *γdR*(*P*(15, 3)) ≤ 24 and *γdR*(*P*(40, 8)) ≤ 64.


**Table 5.** A DRDF of *Ui* that implies *γdR*(*P*(15, 3)) ≤ 24 and *γdR*(*P*(40, 8)) ≤ 64.

*3.2. Double Roman Domination in P*(5*k*, *k*)*—Upper Bounds*

Observe that the patterns used in Tables 4 and 5 have period five (columns). This implies the next proposition.

**Proposition 5.** *Let k* ≡ 2 mod 5 *or k* ≡ 3 mod 5*. Then γdR*(*P*(5*k*, *k*)) ≤ 8*k.*

**Proof.** Recall from previous considerations (Table 4) that *γdR*(*P*(10, 2)) ≤ 16 and *γdR*(*P*(35, 7)) ≤ 56. Observe that if we repeat the columns 2-6 in Table 4, we obtain a DRDF showing *γdR*(*P*(60, 12)) ≤ 96. By induction, it follows that *γdR*(*P*(5(5*i* + 2),(5*i* + 2)) ≤ 8(5*i* + 2) for all integers *i* ≥ 0. Thus, *γdR*(*P*(5*k*, *k*)) ≤ 8*k* for *k* ≡ 2 mod 5.

The statement for *k* ≡ 3 mod 5 follows from Table 5 by analogous argument.

The next table provides a DRDF for *P*(25, 5). It is obtained from Table 4 by deleting columns 3 and 4 and altering only one entry in the original column 5, see Table 6.


**Table 6.** A DRDF of *Ui* that implies *γdR*(*P*(25, 5)) ≤ 42.

It is straightforward to check that the table provides a DRDF. Observe that we can in this way delete two columns and alter one column to obtain a DRDF of *γdR*(*P*(5(5*i*), 5*i*) ≤ 8(5*i*) + 2 from *γdR*(*P*(5(5*i* + 2), 5*i* + 2) ≤ 8(5*i* + 2) for all integers *i* ≥ 0.

The same idea, applied to Table 5 gives a DRDF of *P*(30, 6) of weight 50. We omit the details. Using the periodicity of the basic pattern, we have a construction that gives RDF showing *γdR*(*P*(5(5*i* + 1), 5*i* + 1) ≤ 8(5*i* + 1) + 2 from *γdR*(*P*(5(5*i* + 3),(5*i* + 3)) ≤ 8(5*i* + 3) for all integers *i* ≥ 0.

Similarly, inserting two columns in the pattern comes with additional cost of 8 + 8 + 2 = 18 legions, thus increasing the total weight by 18. For example, see Table 7.


**Table 7.** An alternative DRDF of *Ui* that shows *γdR*(*P*(25, 5)) ≤ 42 and *γdR*(*P*(50, 10)) ≤ 82.

For completeness, in Table 8 we give a RDF of *P*(20, 4) proving that *γdR*(*P*(20, 4)) ≤ 34. The construction starts with RDF of weight 16 for *γdR*(*P*(10, 2)), and inserts two columns as in Table 7. In more detail, note that column 4 is a copy of column 2 and column 3 is a copy of column 1. Then, additional two legions are assigned to vertex *u*<sup>17</sup> in column 1. It follows that *γdR*(*P*(5(5*i* + 4),(5*i* + 4)) ≤ 8(5*i* + 4)+2 for all integers *i* ≥ 0.

**Table 8.** A DRDF of *Ui* that implies *γdR*(*P*(20, 4)) ≤ 34.


Summarizing the arguments, we have a proof of the next proposition.

**Proposition 6.** *Let k* ≡ 0, 1, 4 mod 5*. Then γdR*(*P*(5*k*, *k*)) ≤ 8*k* + 2*.*

#### *3.3. Double Roman Domination in P*(5*k*, *k*)*—Lower Bound*

The proof of lower bound in Theorem 8 is based on several technical lemmas. In all proofs below we assume that *f* is a DRDF and there are no vertices with *f*(*v*) = 1. As before, let *Ui* = {*ui*, *ui*<sup>+</sup>*k*, *ui*<sup>+</sup>2*k*, *ui*<sup>+</sup>3*k*, *ui*<sup>+</sup>4*k*} and *Vi* = {*vi*, *vi*<sup>+</sup>*k*, *vi*<sup>+</sup>2*k*, *vi*<sup>+</sup>3*k*, *vi*<sup>+</sup>4*k*}. Let us denote with *Wi* the weight of *Hi* = *Vi* ∪ *Ui*, *Wi* = *f*(*Hi*) = *f*(*Vi* ∪ *Ui*).

**Lemma 1.** *Let f be DRDF f . Then*

*if f*(*Ui*) = 0 *then f*(*Vi*) ≥ 6 *and Wi* ≥ 6*, if f*(*Ui*) = 2 *then f*(*Vi*) ≥ 5 *and Wi* ≥ 7*, if f*(*Ui*) = 3 *then f*(*Vi*) ≥ 5 *and Wi* ≥ 8*, if f*(*Ui*) = 4 *then f*(*Vi*) ≥ 4 *and Wi* ≥ 8*, if f*(*Ui*) = 5 *then f*(*Vi*) ≥ 4 *and Wi* ≥ 9*, if f*(*Ui*) = 6 *then f*(*Vi*) ≥ 3 *and Wi* ≥ 9*, if f*(*Ui*) = 7 *then f*(*Vi*) ≥ 4 *and Wi* ≥ 11*, if f*(*Ui*) = 8 *then f*(*Vi*) ≥ 3 *and Wi* ≥ 11*, and if f*(*Ui*) ≥ 9 *then Wi* ≥ 12*.*

**Proof.** We will list all possible examples (up to the isomorphism), using the following notation:

 *f*(*ui*) *f*(*ui*<sup>+</sup>*k*) *f*(*ui*<sup>+</sup>2*k*) *f*(*ui*<sup>+</sup>3*k*) *f*(*ui*<sup>+</sup>4*k*) *f*(*vi*) *f*(*vi*<sup>+</sup>*k*) *f*(*vi*<sup>+</sup>2*k*) *f*(*vi*<sup>+</sup>3*k*) *f*(*vi*<sup>+</sup>4*k*) .

• Case *f*(*Ui*) ≥ 9. First, assume *f*(*Ui*) = 9. Excluding weights 1, the sum 9 can be achieved as 9 = 3 + 3 + 3 or 9 = 3 + 2 + 2 + 2. In the first case, three vertices among five can be chosen in two ways, either the two zeros are at adjacent columns or not. Similarly, in the second case, the 0 can either be next to 3, or not. Thus, we have 4 cases listed below. The values on *Vi* are chosen so that the total weight is minimal.

$$
\begin{pmatrix} 3 & 3 & 3 & 0 & 0 \\ 0 & 0 & 0 & 3 & 0 \end{pmatrix}, \quad \begin{pmatrix} 3 & 3 & 0 & 3 & 0 \\ 0 & 0 & 0 & 3 & 0 \end{pmatrix}, \quad \begin{pmatrix} 3 & 2 & 2 & 2 & 0 \\ 0 & 0 & 2 & 0 & 2 \end{pmatrix}, \quad \begin{pmatrix} 3 & 2 & 2 & 0 & 2 \\ 0 & 1 & 0 & 2 & 0 \end{pmatrix}.
$$

In all cases we have *f*(*Vi*) ≥ 3, thus *Wi* ≥ 9 + 3 = 12.

Furthermore, if *f*(*Ui*) = 10 or *f*(*Ui*) = 11 then observe that *f*(*Vi*) ≥ 2, and hence *Wi* ≥ 12.

• Case *f*(*Ui*) = 8. Possible subcases with 8 = 3 + 3 + 2 = 2 + 2 + 2 + 2 are

$$
\begin{pmatrix} 3 & 3 & 2 & 0 & 0 \\ 0 & 0 & 0 & 3 & 0 \end{pmatrix}, \quad \begin{pmatrix} 3 & 3 & 0 & 2 & 0 \\ 0 & 0 & 0 & 3 & 0 \end{pmatrix}, \quad \begin{pmatrix} 3 & 2 & 3 & 0 & 0 \\ 0 & 0 & 2 & 0 & 2 \end{pmatrix}, \quad \begin{pmatrix} 3 & 2 & 0 & 3 & 0 \\ 0 & 0 & 2 & 0 & 2 \end{pmatrix}.
$$

and

$$
\begin{pmatrix} 2 & 2 & 2 & 2 & 0 \\ 0 & 2 & 0 & 0 & 2 \end{pmatrix}.
$$

In all cases, *f*(*Vi*) ≥ 3, thus *Wi* ≥ 8 + 3 = 11.

• Case *f*(*Ui*) = 7. There is only one possibility, 7 = 3 + 2 + 2, and we have the following subcases:

$$
\begin{pmatrix} 3 & 2 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 & 2 \end{pmatrix}, \quad \begin{pmatrix} 3 & 2 & 0 & 2 & 0 \\ 0 & 0 & 2 & 0 & 2 \end{pmatrix}, \quad \begin{pmatrix} 3 & 2 & 0 & 0 & 2 \\ 0 & 0 & 2 & 2 & 0 \end{pmatrix}, \quad \begin{pmatrix} 3 & 0 & 2 & 2 & 0 \\ 0 & 2 & 0 & 0 & 2 \end{pmatrix}.
$$

It is obvious that *f*(*Vi*) ≥ 4 and *Wi* ≥ 7 + 4 = 11.

• Case *f*(*Ui*) = 6. This sum can be achieved as 6 = 3 + 3 = 2 + 2 + 2. There are four subcases:

$$
\begin{pmatrix} 3 & 3 & 0 & 0 & 0 \\ 0 & 0 & 0 & 3 & 0 \end{pmatrix}, \quad \begin{pmatrix} 3 & 0 & 3 & 0 & 0 \\ 0 & 2 & 0 & 0 & 3 \end{pmatrix}, \quad \begin{pmatrix} 2 & 2 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 & 2 \end{pmatrix}, \quad \begin{pmatrix} 2 & 2 & 0 & 2 & 0 \\ 0 & 0 & 2 & 0 & 2 \end{pmatrix}.
$$

In all cases the value *f*(*Vi*) is at least 3, which implies *Wi* ≥ 6 + 3 = 9. • Case *f*(*Ui*) = 5. We have 5 = 3 + 2, and two possibilities.

> 32000 00202 , 30200 02030 .

Clearly, in both cases *f*(*Vi*) must be at least 4, which implies *Wi* ≥ 5 + 4 = 9. • Case *f*(*Ui*) = 4. As 4 = 2 + 2, we have two cases:

> 22000 00202 , 20200 02030 .

As *f*(*Vi*) ≥ 4 in both cases, we have *Wi* ≥ 4 + 4 = 8.


 00000 03003 , 00000 22020 ,

with *f*(*Vi*) = 6, thus *Wi* ≥ 0 + 6 = 6.

This concludes the proof of lemma.

In order to prove the lower bound in Theorem 8, we will need to consider the *Hi* with *Wi* < 8, thus by Lemma 1, the cases *Wi* = 7 (*f*(*Ui*) = 2, and *f*(*Ui*) = 0) or *Wi* = 6 (*f*(*Ui*) = 0). In the Figure 2 below all cases (up to the isomorphism) with *Wi* = 6 and *Wi* = 7 are drawn.

**Figure 2.** The standard drawing of *Hi* (**left**) and the case *f*(*Hi*) = *Wi* = 7 with *f*(*Ui*) = 2 (**right**).

First we consider the cases where two adjacent *Hi* have weights less than 8. Note that the proof of the next lemma also implies that it is not possible to have a DRDF with more that two consecutive *Wi* < 8.

#### **Lemma 2.**


**Proof.** The proof will be derived in several steps using the notation introduced in Table 3. We only give the values on the outer cycle, and in addition, in some cases (for sets *Hi* and *Hi*+1) the values on the inner cycles are provided in parenthesis, as **f**(**uj**)(*f*(*vj*)). For other neighbor sets *H*∗, we will assume that the inner cycles *V*<sup>∗</sup> are completed such that the whole assignment is a DRD function. The weights *Wi* are estimated using the results of Lemma 1.


Therefore, we may assume that *f*(*Ui*) = 2 and *f*(*Ui*+1) = 2. The DRDF for *Hi* is in Figure 2 (right). Considering neighbor sets *Hi*−<sup>1</sup> and *Hi*+<sup>2</sup> we have all subcases listed in Tables 10 and 11.

In Tables 10 and 11, we fix DRDF on *Hi* (second column), and consider all possible DRDF with *Wi*+<sup>1</sup> = 7 and *f*(*Ui*+1) = 2 (third column). The first and fourth columns provide the minimal *f* values in *Hi*−<sup>1</sup> and *Hi*+2, respectively. The labeled graph *Hi*+<sup>1</sup> in this case has no symmetries, hence we have to consider five rotations, and in each case two cases due to reflexion. Thus, we have ten cases in total, b1 to b10, outlined in Tables 10 and 11.

**Figure 3.** The two cases with *f*(*Hi*) = *Wi* = 6.

**Figure 4.** The two cases with *f*(*Hi*) = *Wi* = 7 and *f*(*Ui*) = 0.


**Table 9.** Demands for *Ui*−<sup>1</sup> ∪ *Ui*+<sup>1</sup> when *Wi* = 6 or *Wi* = 7 and *<sup>f</sup>*(*Ui*) = 0.

**Table 10.** Subcases of *<sup>f</sup>*(*Ui*−1), *<sup>f</sup>*(*Ui*), *<sup>f</sup>*(*Ui*+1) and *<sup>f</sup>*(*Ui*+2) with *Wi* = 7, *Wi*+<sup>1</sup> = 7 and *<sup>f</sup>*(*Ui*+1) = <sup>2</sup> (first part).


**Table 11.** Subcases of *<sup>f</sup>*(*Ui*−1), *<sup>f</sup>*(*Ui*), *<sup>f</sup>*(*Ui*+1) and *<sup>f</sup>*(*Ui*+2) with *Wi* = 7, *Wi*+<sup>1</sup> = 7 and *<sup>f</sup>*(*Ui*+1) = <sup>2</sup> (second part).


From Tables 10 and 11 using the results of Lemma 1 we can estimate the weights *Wi*: **(b1)** *Wi*−<sup>1</sup> ≥ 8 + 4 = 12, *Wi*+<sup>2</sup> ≥ 8 + 4 = 12, **(b2)** *Wi*−<sup>1</sup> ≥ 6 + 5 = 11, *Wi*+<sup>2</sup> ≥ 7 + 4 = 11, **(b3)** *Wi*−<sup>1</sup> ≥ 7 + 4 = 11, *Wi*+<sup>2</sup> ≥ 8 + 4 = 12, **(b4)** *Wi*−<sup>1</sup> ≥ 8 + 4 = 12, *Wi*+<sup>2</sup> ≥ 7 + 4 = 11, **(b5)** *Wi*−<sup>1</sup> ≥ 8 + 4 = 12, *Wi*+<sup>2</sup> ≥ 6 + 5 = 11, **(b6)** *Wi*−<sup>1</sup> ≥ 8 + 4 = 12, *Wi*+<sup>2</sup> ≥ 8 + 4 = 12, **(b7)** *Wi*−<sup>1</sup> ≥ 6 + 5 = 11, *Wi*+<sup>2</sup> ≥ 7 + 4 = 11, **(b8)** *Wi*−<sup>1</sup> ≥ 7 + 4 = 11, *Wi*+<sup>2</sup> ≥ 8 + 4 = 12, **(b9)** *Wi*−<sup>1</sup> ≥ 8 + 4 = 12, *Wi*+<sup>2</sup> ≥ 7 + 4 = 11, **(b10)** *Wi*−<sup>1</sup> ≥ <sup>8</sup> + <sup>4</sup> = 12, *Wi*+<sup>2</sup> ≥ <sup>8</sup> + <sup>4</sup> = 12.

(c) Case *Wi* = 6 and *Wi*+<sup>1</sup> = 7. First, observe that in the case when *f*(*Ui*+1) = 0, the reasoning in case (a) and (b) implies that *Wi*−<sup>1</sup> ≥ 12 and *Wi*+<sup>2</sup> ≥ 11. So we can assume that *f*(*Ui*+1) = 2. As seen in Figure 3, three vertices in *Vi* could have weights 2 or two of them have weights 3. As we know, one vertex in *Ui*+<sup>1</sup> has weight 3 and the other one (two steps further) has weight 2, see Figure 2. We thus fix the assignment in *Hi* (second column), add all possible assignments in *Hi*+<sup>1</sup> (third column) and write the minimal weights on *Hi*−<sup>1</sup> and *Hi*+<sup>2</sup> in the first and fourth column. As each of the two assignments of *Hi* is reflexion symmetric, it is clear that there are exactly 10 different cases. All possible outcomes for sets *Hi*−<sup>1</sup> ∪ *Hi* ∪ *Hi*+<sup>1</sup> are given below, Tables 12 and 13.

In Table 12, we find all subcases (c1 to c5) when two vertices in *Vi* have weights 3.

**Table 12.** First five subcases of *<sup>f</sup>*(*Ui*−1), *<sup>f</sup>*(*Ui*), *<sup>f</sup>*(*Ui*+1) and *<sup>f</sup>*(*Ui*+2) with *Wi* = 6 and *Wi*+<sup>1</sup> = 7.


In Table 13, we have all subcases (c6 to c10) when three vertices in *Vi* have weights 2.


**Table 13.** Second five subcases of *<sup>f</sup>*(*Ui*−1), *<sup>f</sup>*(*Ui*), *<sup>f</sup>*(*Ui*+1) and *<sup>f</sup>*(*Ui*+2) with *Wi* = 6 and *Wi*+<sup>1</sup> = 7.

Similarly, as in case (b), we can estimate the weights *Wi* from Tables 12 and 13 using the results of Lemma 1: **(c1)** *Wi*−<sup>1</sup> ≥ 8 + 3 = 11, *Wi*+<sup>2</sup> ≥ 8 + 4 = 12, **(c2)** *Wi*−<sup>1</sup> ≥ 9 + 3 = 12, *Wi*+<sup>2</sup> ≥ 8 + 4 = 12, **(c3)** *Wi*−<sup>1</sup> ≥ 8 + 4 = 12, *Wi*+<sup>2</sup> ≥ 8 + 4 = 12, **(c4)** *Wi*−<sup>1</sup> ≥ <sup>8</sup>+<sup>4</sup> = 12, *Wi*+<sup>2</sup> ≥ <sup>8</sup>+<sup>4</sup> = 12, **(c5)** *Wi*−<sup>1</sup> ≥ <sup>9</sup>+<sup>3</sup> = 12, *Wi*+<sup>2</sup> ≥ <sup>8</sup>+<sup>4</sup> = 12, **(c6)** *Wi*−<sup>1</sup> ≥ 10 + 4 = 14, *Wi*+<sup>2</sup> ≥ 8 + 4 = 12, **(c7)** *Wi*−<sup>1</sup> ≥ 10 + 4 = 14, *Wi*+<sup>2</sup> ≥ 8 + 4 = 12, **(c8)** *Wi*−<sup>1</sup> ≥ 11 + 4 = 15, *Wi*+<sup>2</sup> ≥ 8 + 4 = 12, **(c9)** *Wi*−<sup>1</sup> ≥ 10 + 4 = 14, *Wi*+<sup>2</sup> ≥ <sup>8</sup> + <sup>4</sup> = 12, **(c10)** *Wi*−<sup>1</sup> ≥ <sup>11</sup> + <sup>4</sup> = 15, *Wi*+<sup>2</sup> ≥ <sup>8</sup> + <sup>4</sup> = 12.

#### **Lemma 3.**


**Proof.** As in the proof of Lemma 2, we will use the notation introduced in Table 3. We give only the values on the outer cycle, and in some cases the values on the inner cycles are provided in parenthesis, as **f**(**uj**)(*f*(*vj*)). For other neighbor sets *H*<sup>∗</sup> we will assume that the inner cycles *V*<sup>∗</sup> are completed such that the whole assignment is a DRD function.

(a) Case *Wi* = 6 implies *f*(*Ui*) = 0 (Figure 3). Recall that either three vertices in *Vi* have weight 2 or two of them have weight 3, and that Table 9 gives the demands that need to be fulfilled by the neighboring *H*∗. As *Wi*−<sup>1</sup> ≥ 8 and *Wi*+<sup>1</sup> ≥ 8, according to lemma 1, we have *<sup>f</sup>*(*Ui*−1) ≥ 3 and *<sup>f</sup>*(*Ui*+1) ≥ 3. We may also assume that *<sup>f</sup>*(*Ui*−1) ≤ *<sup>f</sup>*(*Ui*+1). Thus, we have *<sup>f</sup>*(*Ui*−1) ∈ {3, 4, 5, 6}, and all cases are analyzed in Tables 14–16. Note that there is only one case for *<sup>f</sup>*(*Ui*−1) = <sup>6</sup> = <sup>2</sup> + <sup>2</sup> + 2 (and *<sup>f</sup>*(*Ui*+1) = <sup>6</sup> = <sup>2</sup> + <sup>2</sup> + 2), because if *<sup>f</sup>*(*Ui*−1) = <sup>6</sup> = <sup>3</sup> + 3 then *<sup>f</sup>*(*Ui*+1) = <sup>3</sup> < 6.

**Table 14.** Subcases of *<sup>f</sup>*(*Ui*−1), *<sup>f</sup>*(*Ui*) and *<sup>f</sup>*(*Ui*+1) with *Wi* = <sup>6</sup> = <sup>3</sup> + 3.


Reading Table 14 we observe that weights are **(A1-1)**<sup>∗</sup> *Wi*−<sup>1</sup> ≥ 3 + 5 = 8, *Wi*+<sup>1</sup> ≥ 6 + 3 = 9, **(A1-2)**<sup>∗</sup> *Wi*−<sup>1</sup> ≥ 3 + 5 = 8, *Wi*+<sup>1</sup> ≥ 6 + 5 = 11, **(A1-3)**<sup>∗</sup> *Wi*−<sup>1</sup> ≥ 5 + 4 = 9, *Wi*+<sup>1</sup> ≥ 5 + 5 = 10, **(A1-4)**<sup>∗</sup> *Wi*−<sup>1</sup> ≥ 4 + 4 = 8, *Wi*+<sup>1</sup> ≥ 7 + 4 = 11, **(A1-5)**<sup>∗</sup> *Wi*−<sup>1</sup> ≥ 5 + 5 = 10, *Wi*+<sup>1</sup> ≥ 5 + 4 = 9, **(A1-6)** *Wi*−<sup>1</sup> ≥ 5 + 5 = 10, *Wi*+<sup>1</sup> ≥ 5 + 5 = 10, **(A1-7)** *Wi*−<sup>1</sup> ≥ 4 + 5 = 9, *Wi*+<sup>1</sup> ≥ 7 + 4 = 11, **(A1-8)** *Wi*−<sup>1</sup> ≥ 6 + 4 = 10, *Wi*+<sup>1</sup> ≥ 6 + 4 = 10, so in cases (A1-6), (A1-7), and (A1-8), *Wi*−<sup>1</sup> + *Wi*+<sup>1</sup> ≥ 20. However, in the first five

cases (labelled with asterisk), *Wi*−<sup>1</sup> + *Wi*+<sup>1</sup> < 20, and therefore we need to consider *Hi*−<sup>2</sup> and *Hi*+<sup>2</sup> to conclude the proof of assertion (a) of Lemma 3, see Table 15.

**Table 15.** Subcases<sup>∗</sup> of *<sup>f</sup>*(*Ui*−2), *<sup>f</sup>*(*Ui*−1), *<sup>f</sup>*(*Ui*), *<sup>f</sup>*(*Ui*+1) and *<sup>f</sup>*(*Ui*+2) with *Wi*−<sup>1</sup> + *Wi*+<sup>1</sup> < 20.


From Table <sup>15</sup> we can estimate weights **(A1-1)**<sup>∗</sup> *Wi*−<sup>2</sup> ≥ <sup>8</sup> + <sup>4</sup> = 12, *Wi*−<sup>1</sup> ≥ <sup>3</sup> + <sup>5</sup> = 8, *Wi*+<sup>1</sup> ≥ 6 + 3 = 9, *Wi*+<sup>2</sup> ≥ 6 + 5 = 11, **(A1-2)**<sup>∗</sup> *Wi*−<sup>2</sup> ≥ 8 + 4 = 12, *Wi*−<sup>1</sup> ≥ 3 + 5 = 8, *Wi*+<sup>1</sup> ≥ <sup>6</sup> + <sup>5</sup> = 11, *Wi*+<sup>2</sup> ≥ <sup>5</sup> + <sup>5</sup> = 10, **(A1-3)**<sup>∗</sup> *Wi*−<sup>2</sup> ≥ <sup>7</sup> + <sup>4</sup> = 11, *Wi*−<sup>1</sup> ≥ <sup>5</sup> + <sup>4</sup> = 9, *Wi*+<sup>1</sup> ≥ <sup>5</sup> + <sup>5</sup> = 10, *Wi*+<sup>2</sup> ≥ <sup>5</sup> + <sup>5</sup> = 10, **(A1-4)**<sup>∗</sup> *Wi*−<sup>2</sup> ≥ <sup>7</sup> + <sup>4</sup> = 11, *Wi*−<sup>1</sup> ≥ <sup>4</sup> + <sup>4</sup> = 8, *Wi*+<sup>1</sup> ≥ <sup>7</sup> + <sup>4</sup> = 11, *Wi*+<sup>2</sup> ≥ <sup>4</sup> + <sup>5</sup> = 9, **(A1-5)**<sup>∗</sup> *Wi*−<sup>2</sup> ≥ <sup>5</sup> + <sup>5</sup> = 10, *Wi*−<sup>1</sup> ≥ <sup>5</sup> + <sup>5</sup> = 10, *Wi*+<sup>1</sup> ≥ 5 + 4 = 9, *Wi*+<sup>2</sup> ≥ 7 + 4 = 11. Here, in all cases, *Wi*−<sup>2</sup> + *Wi*−<sup>1</sup> ≥ 19, *Wi*+<sup>1</sup> + *Wi*+<sup>2</sup> ≥ 19 and *Wi*−<sup>2</sup> + *Wi*−<sup>1</sup> + *Wi*+<sup>1</sup> + *Wi*+<sup>2</sup> ≥ 39.

**Table 16.** Subcases of *<sup>f</sup>*(*Ui*−1), *<sup>f</sup>*(*Ui*) and *<sup>f</sup>*(*Ui*+1) with *Wi* = <sup>6</sup> = <sup>2</sup> + <sup>2</sup> + 2.


Reading Table 16 we observe that **(A2-1)** *Wi*−<sup>1</sup> ≥ 6 + 5 = 11, *Wi*+<sup>1</sup> ≥ 6 + 4 = 10, **(A2-2)** *Wi*−<sup>1</sup> ≥ 6 + 4 = 10, *Wi*+<sup>1</sup> ≥ 8 + 4 = 12, **(A2-3)** *Wi*−<sup>1</sup> ≥ 6 + 4 = 10, *Wi*+<sup>1</sup> ≥ 8 + 4 = 12, **(A2-4)** *Wi*−<sup>1</sup> ≥ 4 + 5 = 9, *Wi*+<sup>1</sup> ≥ 10 + 4 = 14, **(A2-5)** *Wi*−<sup>1</sup> ≥ 5 + 5 = 10, *Wi*+<sup>1</sup> ≥ 8 + 4 = 12, **(A2-6)** *Wi*−<sup>1</sup> ≥ 5 + 4 = 9, *Wi*+<sup>1</sup> ≥ 7 + 4 = 11, **(A2-7)** *Wi*−<sup>1</sup> ≥ 5 + 5 = 10, *Wi*+<sup>1</sup> ≥ 7 + 4 = 11, **(A2-8)** *Wi*−<sup>1</sup> ≥ 3 + 5 = 8, *Wi*+<sup>1</sup> ≥ 9 + 3 = 12, **(A2-9)** *Wi*−<sup>1</sup> ≥ <sup>6</sup> + <sup>4</sup> = 10, *Wi*+<sup>1</sup> ≥ <sup>7</sup> + <sup>4</sup> = 11, **(A2-10)** *Wi*−<sup>1</sup> ≥ <sup>6</sup> + <sup>4</sup> = 10, *Wi*+<sup>1</sup> ≥ <sup>7</sup> + <sup>4</sup> = 11, **(A2-11)** *Wi*−<sup>1</sup> ≥ 4 + 4 = 8, *Wi*+<sup>1</sup> ≥ 9 + 4 = 13, **(A2-12)** *Wi*−<sup>1</sup> ≥ 4 + 5 = 9, *Wi*+<sup>1</sup> ≥ <sup>9</sup> + <sup>4</sup> = 13, **(A2-13)** *Wi*−<sup>1</sup> ≥ <sup>6</sup> + <sup>4</sup> = 10, *Wi*+<sup>1</sup> ≥ <sup>6</sup> + <sup>5</sup> = 11, **(A2-14)** *Wi*−<sup>1</sup> ≥ <sup>4</sup> + <sup>5</sup> = 9, *Wi*+<sup>1</sup> ≥ 8 + 4 = 12, **(A2-15)** *Wi*−<sup>1</sup> ≥ 4 + 4 = 8, *Wi*+<sup>1</sup> ≥ 8 + 4 = 12, so in all cases *Wi*−<sup>1</sup> + *Wi*+<sup>1</sup> ≥ 20, which proves the assertion (a) of Lemma 3.

(b) Case *Wi* = 7. First, assume that *f*(*Ui*) = 2 (see Figure 2), so one vertex in *Ui* has weight 3 and the other one has weight 2. Possible (due to symmetry) solutions for the whole set *Hi*−<sup>1</sup> ∪ *Hi* ∪ *Hi*+<sup>1</sup> are considered in the following Table 17.


**Table 17.** Possible values for *<sup>f</sup>*(*Ui*−1), *<sup>f</sup>*(*Ui*) and *<sup>f</sup>*(*Ui*+1) with *Wi* = 7 and *<sup>f</sup>*(*Ui*) = 2.

Without loss of generality, assume that *<sup>f</sup>*(*Ui*−1) ≤ *<sup>f</sup>*(*Ui*+1). As *Wi*−<sup>1</sup> ≥ 8 and *Wi*+<sup>1</sup> ≥ <sup>8</sup> we have the following subcases (see Table 18).

**Table 18.** Subcases of *<sup>f</sup>*(*Ui*−1), *<sup>f</sup>*(*Ui*) and *<sup>f</sup>*(*Ui*+1) with *Wi* = 7, *<sup>f</sup>*(*Ui*) = 2, *Wi*−<sup>1</sup> ≥ 8 and *Wi*+<sup>1</sup> ≥ 8.


Reading Table <sup>18</sup> we observe that in all cases except one (B1-4) we have *Wi*−<sup>1</sup> + *Wi*+<sup>1</sup> ≥ 18. Indeed, in **(B1-1)** *Wi*−<sup>1</sup> ≥ 5 + 5 = 10, *Wi*+<sup>1</sup> ≥ 3 + 5 = 8, **(B1-2)** *Wi*−<sup>1</sup> ≥ 4 + 5 = 9, *Wi*+<sup>1</sup> ≥ 5 + 5 = 10, **(B1-3)** *Wi*−<sup>1</sup> ≥ 4 + 4 = 8, *Wi*+<sup>1</sup> ≥ 5 + 5 = 10, **(B1-4)**<sup>∗</sup> *Wi*−<sup>1</sup> ≥ 3 + 5 = 8, *Wi*+<sup>1</sup> ≥ 5 + 4 = 9, **(B1-5)** *Wi*−<sup>1</sup> ≥ 3 + 5 = 8, *Wi*+<sup>1</sup> ≥ 5 + 5 = 10, **(B1-6)** *Wi*−<sup>1</sup> ≥ 4 + 5 = 9, *Wi*+<sup>1</sup> ≥ 6 + 4 = 10. In case (**B1-4**) we need to consider weights on *Hi*−<sup>2</sup> and *Hi*+<sup>2</sup> to conclude the proof of assertion (b) of Lemma 3.

As seen from Table 19, in case (**B1-4**) we can confirm *Wi*−<sup>2</sup> + *Wi*−<sup>1</sup> ≥ 18 and *Wi*+<sup>1</sup> + *Wi*+<sup>2</sup> ≥ 18.


**Table 19.** Subcase **(B1-4)** with *Wi*−<sup>2</sup> ≥ 7 + 4 = 11, *Wi*−<sup>1</sup> ≥ 3 + 5 = 8, *Wi*+<sup>1</sup> ≥ 5 + 4 = 9 and *Wi*+<sup>2</sup> ≥ 5 + 4 = 9.

Finally, if *f*(*Ui*) = 0 (see Figure 4), then observe that in comparison to the case *<sup>f</sup>*(*Ui*) = 2, there must be at least one more vertex of weight 2 in *Ui*−<sup>1</sup> ∪ *Ui*+1. We omit detailed analysis of the cases that confirm *Wi*−<sup>1</sup> + *Wi*+<sup>1</sup> ≥ 18.

**Theorem 8.** *For all k we have γdR*(*P*(5*k*, *k*)) ≥ 8*k*.

**Proof.** We use the discharging method (see [14]). The basic idea is as follows. Assume that we have a DRDF. Consider certain subgraphs, in our case the subgraphs *Hi*, that are induced on *Vi* ∪ *Ui*. Define some rules how the weights of heavy subgraphs are discharged to the neighbors such that the total weight does not change. Observe the weights of subgraphs after discharging.

In our case, the discharging rule is simple: If *f*(*Hi*) = *Wi* > 8 then *Hi* sends <sup>1</sup> <sup>2</sup> (*Wi* − 8) to *Hi*−*<sup>i</sup>* and to *Hi*−*i*. The new charge of *Hi* is thus 8. We denote the charges after the first round by *W*∗ *i* .

Now we show that if *f*(*Hi*) < 8 then, after at most four rounds of discharging, the new charge *W*∗∗∗∗ *<sup>i</sup>* of *Hi* is at least 8. Note that once the charge of *Hi* is at least 8, discharging will never decrease its charge below 8.

First, let us consider the cases of Lemma 2.


If *Wi*−<sup>1</sup> ≥ 12, *Wi*+<sup>2</sup> ≥ 11, then observe that already *W*<sup>∗</sup> *<sup>i</sup>* ≥ 8 and *W*<sup>∗</sup> *<sup>i</sup>*+<sup>1</sup> ≥ 8.

Next, we consider the cases of Lemma 3.

	- (a11) *Wi*−<sup>1</sup> <sup>=</sup> 9 and *Wi*−<sup>2</sup> <sup>=</sup> 10. In the first round of discharging, *Hi* receives <sup>1</sup> <sup>2</sup> from *Hi*−1, and *W*<sup>∗</sup> *<sup>i</sup>*−<sup>1</sup> <sup>=</sup> <sup>9</sup> <sup>−</sup> <sup>2</sup> · <sup>1</sup> <sup>2</sup> + 1 = 9. In the second round of discharging, *Hi* again receives <sup>1</sup> <sup>2</sup> from *Hi*−1, and in total *Hi* receives charge 1 from the left side.
	- (a12) *Wi*−<sup>1</sup> = 8 and *Wi*−<sup>2</sup> = 11. After the first round of discharging, *<sup>W</sup>*<sup>∗</sup> *<sup>i</sup>*−<sup>1</sup> <sup>=</sup> <sup>8</sup> <sup>+</sup> <sup>3</sup> <sup>2</sup> . In the second round of discharging, *Hi*−<sup>1</sup> sends <sup>3</sup> <sup>4</sup> to its neighbors. Thus, *Hi* receives 3 <sup>4</sup> from the left side.

Recall that by Lemma 3, *Wi*−<sup>2</sup> + *Wi*−<sup>1</sup> = 19 implies *Wi*+<sup>1</sup> + *Wi*+<sup>2</sup> ≥ 20, and distinguish two cases.


Summarizing, *Hi* receives charge at least <sup>3</sup> <sup>4</sup> from the left side, and at least <sup>5</sup> <sup>4</sup> from the right side. Hence *W*∗∗∗∗ *<sup>i</sup>* ≥ 6 + 2 = 8, as claimed.

	- (b1) *Wi*−<sup>1</sup> <sup>=</sup> 9. In the first round of discharging, *Hi* receives <sup>1</sup> <sup>2</sup> from the left side.
	- (b2) If *Wi*−<sup>1</sup> = 8, then *Wi*−<sup>2</sup> ≥ 10. After the first round of discharging, *<sup>W</sup>*<sup>∗</sup> *<sup>i</sup>*−<sup>1</sup> <sup>≥</sup> <sup>8</sup> <sup>+</sup> 1. In the second round of discharging, *Hi*−<sup>1</sup> sends at least <sup>1</sup> <sup>2</sup> to its neighbors.

Thus, *Hi* receives at least <sup>1</sup> <sup>2</sup> from the left side. By analogous reasoning, *Wi*+<sup>1</sup> + *Wi*+<sup>2</sup> ≥ 18 implies that *Hi* receives at least <sup>1</sup> <sup>2</sup> from the right side. Consequently, in total *Hi* receives at least <sup>1</sup> <sup>2</sup> <sup>+</sup> <sup>1</sup> <sup>2</sup> , as claimed.

Therefore, after discharging, each subgraph *Hi* has weight *Wi* at least 8, and consequently, the total weight is at least 8*k*, as claimed.

#### **4. Domination in Generalized Petersen Graphs** *P***(20***k***,** *k***)**

In this section, we discuss how the constructions, and the corresponding upper bounds can be extended from *P*(*c*0*k*, *k*) to *P*((*hc*0)*k*, *k*), for *h* = 2, 3, ... . In particular, we obtain improved upper bounds for *P*(20*k*, *k*) from upper bounds for *P*(4*k*, *k*) and *P*(5*k*, *k*).

First, we recall the notion of covering graph and *h*-lift to observe that *P*(20*k*, *k*) is a covering graph of both *P*(4*k*, *k*) and *P*(5*k*, *k*). For basic information on covering graphs see [39]. Here we follow the approach used in [40]. Let *G* = (*V*1, *E*1) and *H* = (*V*2, *E*2) be two graphs, and let *p* : *V*<sup>2</sup> → *V*<sup>1</sup> be a surjection. We will call *p* a *covering map* from *H* to *G* if for each *v* ∈ *V*2, the restriction of *p* to the neighborhood of *v* ∈ *V*<sup>2</sup> is a bijection onto the neighborhood of *p*(*v*) in *G*. In other words, *p* maps edges incident to *v* one-to-one onto edges incident to *p*(*v*). *H* is called a *covering graph*, or a lift, of *G* if there exists a covering map from *H* to *G*. Assuming *H* is a lift of *G* with a covering map *p*. If *p* has a property that for every vertex *<sup>v</sup>* <sup>∈</sup> *<sup>V</sup>*(*G*), its *fiber p*−1(*v*) has exactly *<sup>h</sup>* elements, we call *<sup>H</sup>* <sup>a</sup> *<sup>h</sup>*-lift of *<sup>G</sup>*.

Obviously, a long cycle may be a covering graph of shorter cycles. For example, the cycle *C*<sup>100</sup> is a 2-lift of *C*50, considering the surjection *p*(*vi*) = *vi* mod 50. Furthermore, *C*<sup>100</sup> is also a 25-lift of *C*4, etc.

**Proposition 7.** *Let k* ≥ 1*, c*<sup>0</sup> ≥ 3*, and h* ≥ 2*. Petersen graph P*((*hc*0)*k*, *k*) *is a h-lift of P*(*c*0*k*, *k*)*.*

**Proof.** Consider the surjection *p* : *V*(*P*((*hc*0)*k*, *k*)) → *V*(*P*(*c*0*k*, *k*)) defined by *p*(*vi*) = *vi* mod *<sup>h</sup>*, and *p*(*ui*) = *ui* mod *<sup>h</sup>*.

**Proposition 8.** *γdR*(*P*((*hc*0)*k*, *k*)) ≤ *hγdR*(*P*(*c*0*k*, *k*))*.*

**Proof.** Let *f* be a DRDF of *P*(*c*0*k*, *k*). Define ˜ *f* as ˜ *f*(*v*) = *f*(*p*(*v*)) and observe that ˜ *f* is a DRDF of *P*((*hc*0)*k*, *k*). Clearly, the weight ˜ *f*(*P*((*hc*0)*k*, *k*)) is exactly *h* ˜ *f*(*P*((*c*0*k*, *k*)). We omit the details.

**Corollary 3.** *γdR*(*P*(20*k*, *k*)) ≤ 5*γdR*(*P*(4*k*, *k*)) ≤ 30*k* + 15*.*

As *<sup>γ</sup>dR*(*P*(20*k*, *<sup>k</sup>*)) <sup>≥</sup> <sup>3</sup> <sup>2</sup> 20*k* = 30*k* by Corollary 1, we also have

**Corollary 4.** *If k* ≡ 1 mod 2*, then γdR*(*P*(20*k*, *k*)) = 30*k.*

Applying Proposition 8 to the case *P*(20*k*, *k*) and *P*(5*k*, *k*), we obtain another Corollary.

**Corollary 5.** *γdR*(*P*(20*k*, *k*)) ≤ 4*γdR*(*P*(5*k*, *k*)) = 32*k* + 8*.*

Clearly, the upper bound in Corollary 5 is only better for *k* = 1, 2, and 3. In these cases, we obtain

$$
\gamma\_{dR}(P(\text{20},1)) \le 40, \quad \gamma\_{dR}(P(\text{40},2)) \le 64, \quad \gamma\_{dR}(P(\text{60},3)) \le 96,\tag{3}
$$

which, together with Corollary 3 and Theorem 3 implies Theorem 7.

Note that this bound is a considerable improvement over the general bounds given in Theorem 4 [12]. Indeed, the upper bound (2) grows as O(30*k*). The bounds in Theorem 4 are of the from <sup>3</sup> <sup>2</sup> (20*k*)*F*(*k*) + <sup>5</sup>*<sup>k</sup>* <sup>4</sup> + *<sup>C</sup>*, where lim*k*→<sup>∞</sup> *<sup>F</sup>*(*k*) = 1, so the asymptotic growth is <sup>O</sup>(30*<sup>k</sup>* <sup>+</sup> <sup>5</sup>*<sup>k</sup>* <sup>4</sup> ).

#### **5. Conclusions and Future Work**

In this paper, we have extended the known results on double Roman domination of families *P*(*ck*, *k*) of generalized Petersen graphs, by adding either exact values or bounds with gap at most 2 for the family *P*(5*k*, *k*). This naturally continues previous work, where the families *P*(3*k*, *k*) and *P*(4*k*, *k*) were studied.

There are several interesting related questions that open avenues for future work. For example:


The authors believe that this study has solved the problem on *P*(5*k*, *k*) to the limits of the standard method. These methods may be sufficient to handle the problem, e.g., on *P*(6*k*, *k*), but probably can not be applied to much larger *c*. Covering graphs, as indicated in Section 4, may provide a tool to provide improved bounds for larger *c*. On the other hand, the gaps between the lower and upper bounds in some cases may be solved by other methods, see for example [41] and the references there.

More generally, this work again shows the power of the discharging method. The discharging method is most well known for its central role in the proof of the four color theorem. This proof technique was extensively applied to study various graph coloring problems, in particular on planar graphs. In [14], it is shown that a suitably altered discharging technique can also be used on domination type problems and is illustrated on the double Roman domination on some generalized Petersen graphs. Here, we apply the method to another family of graphs and the same problem. This may encourage future applications to other domination type problems.

**Author Contributions:** D.R.P. and J.Ž. contributed equally to this work. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported in part by the Slovenian Research Agency ARRS (grants P2-0248, J2-2512, and J1-1693).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The authors wish to sincerely thank to the anonymous reviewers for their constructive comments.

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

#### **References**

