# Perfect domination, Roman domination and perfect Roman domination in lexicographic product graphs

The aim of this paper is to obtain closed formulas for the perfect domination number, the Roman domination number and the perfect Roman domination number of lexicographic product graphs. We show that these formulas can be obtained relatively easily for the case of the first two parameters. The picture is quite different when it concerns the perfect Roman domination number. In this case, we obtain general bounds and then we give sufficient and/or necessary conditions for the bounds to be achieved. We also discuss the case of perfect Roman graphs and we characterize the lexicographic product graphs where the perfect Roman domination number equals the Roman domination number.

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## 1 Introduction

Given a graph , a set of vertices is a dominating set if every vertex in is adjacent to at least one vertex in . Let be the set of dominating sets of . The domination number of is defined to be,

 γ(G)=min{|S|:S∈D(G)}.

Now, is a perfect dominating set of if every vertex in is adjacent to exactly one vertex in . Let be the set of perfect dominating sets of . The perfect domination number of is defined to be,

 γp(G)=min{|S|:S∈Dp(G)}.

Notice that , which implies that .

The domination number has been extensively studied. For instance, we cite the following books, [17, 18]. The theory of perfect domination was introduced by Livingston and Stout in [26] and has been studied by several authors, including [9, 11, 13, 15, 22, 24].

Cockayne, Hedetniemi and Hedetniemi [10] defined a Roman dominating function, abbreviated RDF, on a graph to be a function satisfying the condition that every vertex for which is adjacent to at least one vertex for which . The weight of is defined to be

 ω(f)=∑v∈V(G)f(v).

For we define the weight of as . The Roman domination number, denoted by , is the minimum weight among all Roman dominating functions on , i.e.,

 γR(G)=min{ω(f):f is an RDF on G}.

An RDF of weight is called a -function. Obviously, for every graph . A Roman graph is a graph with .

Recently, a perfect version of Roman domination was introduced by Henning, Klostermeyer and MacGillivray [20]. They defined a perfect Roman dominating function, abbreviated PRDF, as an RDF satisfying the condition that every vertex for which is adjacent to exactly one vertex for which . The perfect Roman domination number, denoted by , is the minimum weight among all perfect Roman dominating functions on , i.e.,

 γpR(G)=min{ω(f):f is a PRDF on G}.

For results on perfect Roman domination in graphs we cite [3, 12, 19, 33].

A PRDF of weight is called a -function. Observe that for every graph . Those graphs attaining the equality are called perfect Roman graphs. All perfect Roman trees were characterized in [29].

Figure 1 shows three copies of a graph with . Notice that the labellings correspond to the positive weights of all -functions. In particular, the labellings on the center and on the right correspond to the positive weights of -functions.

Figure 2 shows a Roman graph , namely, . In this case, and . The set of labelled vertices form a -set and the labels describe the positive weights of a -function.

The aim of this paper is to obtain closed formulas for the perfect domination number, the Roman domination number and the perfect Roman domination number of lexicographic product graphs. The paper is organised as follows. In Section 2 we declare the general notation, terminology and basic tools needed to develop the remaining sections. In Section 3 we obtain closed formulas for the perfect domination number and the Roman domination number of lexicographic product graphs. Finally, Section 4 is devoted to provide tight bounds and closed formulas for the perfect Roman domination number of lexicographic product graphs.

## 2 Notation, terminology and basic tools

Throughout the paper, we will use the notation and for a complete graph and an empty graph of order , respectively. We use the notation if and are adjacent vertices, and if and are isomorphic graphs. For a vertex of a graph , will denote the set of neighbours or open neighbourhood of , i.e., . The closed neighbourhood, denoted by , equals . Given a set and a vertex , the external private neighbourhood of with respect to is defined to be .

We denote by the degree of vertex , as well as the minimum degree of , the maximum degree of and the order of . Given a set , , and the subgraph of induced by will be denoted by .

A set is a total dominating set of a graph without isolated vertices if every vertex is adjacent to at least one vertex in . Let be the set of total dominating sets of . The total domination number of is defined to be,

 γt(G)=min{|S|:S∈Dt(G)}.

By definition, , so that . Furthermore, . We define a -set as a set with . The same agreement will be assumed for optimal parameters associated to other characteristic sets defined in the paper. For instance, a -set will be a set with .

A graph invariant closely related to the domination number is the packing number. A set is a packing if for every pair of different vertices . We define

 ℘(G)={S⊆V(G):S is a packing of G}.

The packing number, denoted by , is the maximum cardinality among all packings of , i.e.,

 ρ(G)=max{|S|:S∈℘(G)}.

Obviously, . Furthermore, Meir and Moon [27] showed in 1975 that for every tree . We would point out that, in general, these -sets and -sets are not identical. Notice that if and only if there exists a -set which is a -set. A graph is an efficient closed domination graph if .

A set is an open packing, if for every pair of different vertices . We define

 ℘o(G)={S⊆V(G):S is an open packing of G}.

The open packing number of , denoted by , is the maximum cardinality among all open packings of , i.e.,

 ρo(G)=max{|S|:S∈℘o(G)}.

By definition, , so that for every graph , and for every graph without isolated vertices. Besides, if , then every vertex of has degree at most one, which implies that we can write , where is the set of isolated vertices of and . Obviously, if and only if .

A graph is an efficient open domination graph if there exists a set , called an efficient open dominating set, for which and for every pair of distinct vertices . As shown in [23], if is an efficient open domination graph with an efficient open dominating set , then . Hence, the following remark holds.

###### Remark 2.1.

A graph is an efficient open domination graph if and only if there exists such that . In such a case, .

###### Corollary 2.2.

If is an efficient open domination graph, then .

Given two nontrivial graphs and , we define the following properties, which will become important tools in the next sections.

1. :   and is an efficient open domination graph.

2. :   and is an efficient closed domination graph.

3. :   , is an efficient open domination graph and .

Let be a function on and let , where . We will identify with the subsets , and so we will use the unified notation for the function and these associated subsets.

An RDF on is a total Roman dominating function if [1]. The total Roman domination number, denoted by , is the minimum weight among all total Roman dominating functions on . By definition, .

The lexicographic product of two graphs and is the graph whose vertex set is and if and only if or and . For simplicity, the neighbourhood of will be denoted by instead of , and for any PRDF on we will write instead of .

Notice that for any the subgraph of induced by is isomorphic to . We will denote this subgraph by . For any and any function on we define

 f(Hu)=∑v∈V(H)f(u,v) and f[Hu]=∑x∈N[u]f(Hx).

For basic properties of the lexicographic product of two graphs we suggest the books [16, 21]. A main problem in the study of product of graphs consists of finding exact values or sharp bounds for specific parameters of the product of two graphs and express them in terms of invariants of the factor graphs. In particular, we cite the following works on domination theory of lexicographic product graphs. For instance, the reader is referred to [25, 28] for the domination number, [4] for the double domination number, [30] for the Roman domination number, [6, 8] for the total Roman domination number, [31] for the rainbow domination number, [14] for the super domination number, [32] for the weak Roman domination number, [7] for the total weak Roman domination number and the secure total domination number, [5] for the Italian domination number and [2] for the doubly connected domination number.

For the remainder of the paper, definitions will be introduced whenever a concept is needed.

## 3 Perfect domination and Roman Domination in lexicographic product graphs

The next theorem merges two results obtained in [30] and [34].

###### Theorem 3.1 ([30] and [34]).

For any graph with no isolated vertex and any nontrivial graph ,

 γ(G∘H)={\parγ(G)if γ(H)=1,\parγt(G)if γ(H)≥2.\par

As the following result shows, when computing the perfect domination number of lexicographic product graphs , where is connected and is not trivial, we have to take into account that the class of graphs satisfies a certain trichotomy, as it is divided into three categories, i.e., the class of graphs for which holds, the class of graphs for which holds, and the class where nor neither holds.

###### Theorem 3.2.

For any connected graph and any nontrivial graph ,

 γp(G∘H)=⎧⎪⎨⎪⎩\parγt(G) if P1(G,H) holds,\parγ(G) if P2(G,H) holds,\parn(G)n(H) otherwise.\par
###### Proof.

Let be a -set and define and . We differentiate, the following two cases.

Case 1. There exists such that . Since , we deduce that , which implies that , i.e., .

Case 2. for every . Obviously, and, since for every , we conclude that . Let . If is an isolated vertex of , then is a universal vertex of , while if has degree one, then is an isolated vertex of . Therefore, we have the following two complementary subcases.

Subcase 2.1. holds, i.e., is an isolated vertex of , and . In this case, Remark 2.1 leads to . Hence, .

Subcase 2.2. holds, i.e., is a universal vertex of , is -set and also a -set. In this case, . ∎

The Roman domination number of the lexicographic product of two connected graphs and was studied in [30]. Obviously, the connectivity of only depends on the connectivity of . Since we need to consider the case where is not necessarily connected, we make next the necessary modifications to adapt the results obtained in [30] to the general case in which is not necessarily connected.

###### Lemma 3.3.

Let be a graph with no isolated vertex and a nontrivial graph. Let be a -function, and . If is maximum among all -functions, then and .

###### Proof.

Let be a -function such that is maximum among all -functions. If , then , which implies that . Hence, .

Now, suppose that there exists . Observe that , and so . Given and , we define a function on by , and for the remaining vertices. Notice that is a RDF on with and, since is a nontrivial graph, , so that , which is a contradiction. Therefore, and . ∎

The following result is a direct consequence of Lemma 3.3.

###### Corollary 3.4.

For any graph without isolated vertices and any nontrivial graph ,

 γR(G∘H)≥2γ(G).
###### Theorem 3.5.

[30] For any graph without isolated vertices and any graph ,

 γR(G∘H)≤2γt(G).

Now, we introduce the definition of domination couple given in [30]. We say that an ordered couple of disjoint sets is a dominating couple of if every vertex satisfies that . Also, we define the parameter as follows.

 ζ(G)=min{2|A|+3|B|:(A,B) is a dominating couple of G}.

We say that a dominating couple of is a -couple if . With this notation in mind, we state the following result.

###### Theorem 3.6.

For any graph without isolated vertices and any nontrivial graph ,

 γR(G∘H)=⎧⎪⎨⎪⎩2γ(G)if Δ(H)=n(H)−1,ζ(G)if Δ(H)=n(H)−2,2γt(G)if Δ(H)≤n(H)−3.
###### Proof.

As shown in [30], if and is a connected nontrivial graph, then Obviously, the same equality holds if is not connected.

In order to discuss the remaining cases, let be a -function such that is maximum. By Lemma 3.3, is a dominating set of and . Let .

Assume . Since is a dominating couple of , we deduce that . Now, let be a vertex of maximum degree and . Since for any -couple , the function , defined by and , is an RDF on , we deduce that . Therefore, .

Finally, assume . By Theorem 3.5 we only need to prove that In this case, if , then , while if , then . Since does not have isolated vertices and , we have that . Hence, , which completes the proof. ∎

Two simple characterizations of Roman graphs were given in [10], but the authors suggest finding classes of Roman graphs. The following result is an immediate consequence of Theorems 3.1 and 3.6.

###### Theorem 3.7.

Let be a graph with no isolated vertex. If is a graph such that , then is a Roman graph.

As has not been extensively studied, we next obtain tight bounds on for the case in which .

###### Theorem 3.8.

Let a graph with no isolated vertex and a graph. If , then

 max{γtR(G),γt(G)+γ(G)}≤γR(G∘H)≤min{3γ(G),2γt(G)}.
###### Proof.

Let be a -function with maximum. As above, let , and . By Lemma 3.3, and . Furthermore, if , then , while if , then . Thus,

 γR(G∘H)=3|A′f|+2|Af∖A′f|.

We first prove the lower bounds. Let be a set of minimum cardinality among the sets satisfying that and for every vertex . Since , we deduce that . Hence, .

Now, let be a function on defined by and . Notice that is a TRDF on . Thus, , which completes the proof of the lower bounds.

In order to prove the upper bounds, let be a -set, and let such that is a vertex of maximum degree and . Notice that the function , defined by and , is an RDF on . Therefore, .

Finally, the bound is already known from Theorem 3.5. Therefore, the proof is complete. ∎

The bounds above are tight. Notice that, if , then while if , then we have

## 4 Perfect Roman domination in lexicographic product graphs

This section is organised as follows. First we obtain tight bounds on and then we give sufficient and/or necessary conditions for the bounds to be achieved. We also discuss the case of perfect Roman graphs and we characterize the graphs where .

###### Theorem 4.1.

For any graph without isolated vertices and any graph ,

 γpR(G∘H)≤γp(G)(n(H)+1).
###### Proof.

Let be a -set and . Let be a function on defined by and . Clearly, is a PRDF, which implies that . Therefore, the result follows. ∎

In order to see that the bound above is tight, we can consider the corona graph , where , is any graph of minimum degree at least two, and is a nontrivial graph. In this case,

###### Theorem 4.2.

Let be a graph without isolated vertices and a graph. The following statements hold.

1. For any -function ,

 γpR(G∘H)≤γpR(G)+(|V1|+|V2|)(n(H)−1).
2. If there exists a -function such that is a -set, then

 γpR(G∘H)≤γpR(G)n(H)−γ(G)(n(H)−1).
3. If is a -set, and , then

 γpR(G∘H)≤|S′|+2|S′′|+γp(G)(n(H)−1).
4. If there exists a -function such that is a -set, then

 γpR(G∘H)≤γpR(G)+γp(G)(n(H)−1).
###### Proof.

From any -function , we can define a function on as and . It is readily seen that is a PRDF and, as a result, . Therefore, (i) follows.

Now, since from (i) we deduce (ii).

In order to prove (iii), we only need to observe that for any -set , the function is a PRDF on . Thus, we conclude the proof of (iii) by analogy to the proof of (i), by using instead of .

Finally, (iv) follows from (i). ∎

The bounds above are tight. For instance, let be the graph shown in Figure 2, the set of vertices labelled with , the set of vertices labelled with and . In this case, is a -set, is a -function, is a -set and for every graph . Therefore, the bounds above are achieved.

###### Theorem 4.3.

For any graph without isolated vertices and any graph ,

 γpR(G∘H)≤minS∈℘o(G){|S0|(n(H)−Δ(H)+1)+|S1|(2+δ(H))+n(H)(n(G)−|N[S]|)}.
###### Proof.

Let and such that and . From , and , we can construct a function on as follows. Let and . It is readily seen that is a PRDF on . Therefore, Since the inequality holds for any open packing of , the result follows. ∎

The following result is an immediate consequence of Theorem 4.3.

###### Corollary 4.4.

Given a graph without isolated vertices, the following statements hold.

1. If is an efficient open domination graph, then for any graph ,

 γpR(G∘H)≤γt(G)(2+δ(H)).
2. If is an efficient closed domination graph, then for any graph ,

 γpR(G∘H)≤γ(G)(n(H)−Δ(H)+1).
###### Proof.

First, we proceed to prove (i). Let such that . Notice that and . Hence, by Theorem 4.3 and Remark 2.1 we deduce that