Perfect domination, Roman domination and perfect Roman domination in lexicographic product graphs
A. Cabrera Martinez, C. Garcia-Gomez, J. A. Rodriguez-Velazquez
aa r X i v : . [ c s . D M ] J a n Perfect domination, Roman domination and perfectRoman domination in lexicographic product graphs
A. Cabrera Mart´ınez, C. Garc´ıa-G ´omez, J. A. Rodr´ıguez-Vel´azquez
Universitat Rovira i VirgiliDepartament d’Enginyeria Inform`atica i Matem`atiquesAv. Pa¨ısos Catalans 26, 43007 Tarragona, [email protected], [email protected], [email protected]
January 7, 2021
Abstract
The aim of this paper is to obtain closed formulas for the perfect domination num-ber, the Roman domination number and the perfect Roman domination number of lexi-cographic product graphs. We show that these formulas can be obtained relatively easilyfor the case of the first two parameters. The picture is quite different when it concerns theperfect Roman domination number. In this case, we obtain general bounds and then wegive sufficient and/or necessary conditions for the bounds to be achieved. We also discussthe case of perfect Roman graphs and we characterize the lexicographic product graphswhere the perfect Roman domination number equals the Roman domination number.
Keywords : Roman domination; perfect domination; perfect Roman domination; lexicographicproduct
Given a graph G , a set S ⊆ V ( G ) of vertices is a dominating set if every vertex in V ( G ) \ S is adjacent to at least one vertex in S . Let D ( G ) be the set of dominating sets of G . The domination number of G is defined to be, γ ( G ) = min {| S | : S ∈ D ( G ) } . Now, S ⊆ V ( G ) is a perfect dominating set of G if every vertex in V ( G ) \ S is adjacent toexactly one vertex in S . Let D p ( G ) be the set of perfect dominating sets of G . The perfectdomination number of G is defined to be, γ p ( G ) = min {| S | : S ∈ D p ( G ) } . Notice that D p ( G ) ⊆ D ( G ) , which implies that γ ( G ) ≤ γ p ( G ) .1he domination number has been extensively studied. For instance, we cite the followingbooks, [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 , ab-breviated RDF, on a graph G to be a function f : V ( G ) −→ { , , } satisfying the conditionthat every vertex u for which f ( u ) = v for which f ( v ) = weight of f is defined to be ω ( f ) = ∑ v ∈ V ( G ) f ( v ) . For X ⊆ V ( G ) we define the weight of X as f ( X ) = ∑ v ∈ X f ( v ) . The Roman domination num-ber , denoted by γ R ( G ) , is the minimum weight among all Roman dominating functions on G , i.e. , γ R ( G ) = min { ω ( f ) : f is an RDF on G } . An RDF of weight γ R ( G ) is called a γ R ( G ) -function. Obviously, γ R ( G ) ≤ γ ( G ) for everygraph G . A Roman graph is a graph G with γ R ( G ) = γ ( G ) .Recently, a perfect version of Roman domination was introduced by Henning, Kloster-meyer and MacGillivray [20]. They defined a perfect Roman dominating function , abbreviatedPRDF, as an RDF f satisfying the condition that every vertex u for which f ( u ) = v for which f ( v ) =
2. The perfect Roman domination number , denotedby γ pR ( G ) , is the minimum weight among all perfect Roman dominating functions on G , i.e. , γ pR ( 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 γ pR ( G ) is called a γ pR ( G ) -function. Observe that γ R ( G ) ≤ γ pR ( G ) ≤ γ p ( G ) for every graph G . Those graphs attaining the equality γ pR ( G ) = γ p ( G ) are called perfect Roman graphs . All perfect Roman trees were characterized in [29].Figure 1 shows three copies of a graph G with γ R ( G ) = γ pR ( G ) =
4. Notice that the la-bellings correspond to the positive weights of all γ R ( G ) -functions. In particular, the labellingson the center and on the right correspond to the positive weights of γ pR ( G ) -functions. Figure 1: The labellings associated to the positive weights of all γ R ( G ) -functions on the samegraph. The labellings on the center and on the right correspond to the case of γ pR ( G ) -functions.Figure 2 shows a Roman graph G , namely, γ R ( G ) = = γ ( G ) . In this case, γ p ( G ) = γ pR ( G ) =
9. The set of labelled vertices form a γ p ( G ) -set and the labels describe the positiveweights of a γ pR ( G ) -function.The aim of this paper is to obtain closed formulas for the perfect domination number, theRoman domination number and the perfect Roman domination number of lexicographic prod-uct graphs. The paper is organised as follows. In Section 2 we declare the general notation,2 Figure 2: The set of labelled vertices form a γ p ( G ) -set and the labels correspond to the positiveweights of a γ pR ( G ) -function.terminology and basic tools needed to develop the remaining sections. In Section 3 we obtainclosed formulas for the perfect domination number and the Roman domination number of lex-icographic product graphs. Finally, Section 4 is devoted to provide tight bounds and closedformulas for the perfect Roman domination number of lexicographic product graphs. Throughout the paper, we will use the notation K k and N k for a complete graph and an emptygraph of order k , respectively. We use the notation u ∼ v if u and v are adjacent vertices,and G ∼ = H if G and H are isomorphic graphs. For a vertex v of a graph G , N ( v ) will denotethe set of neighbours or open neighbourhood of v , i.e. , N ( v ) = { u ∈ V ( G ) : u ∼ v } . The closed neighbourhood , denoted by N [ v ] , equals N ( v ) ∪ { v } . Given a set S ⊆ V ( G ) and a vertex v ∈ S , the external private neighbourhood epn ( v , S ) of v with respect to S is defined to beepn ( v , S ) = { u ∈ V ( G ) \ S : N ( u ) ∩ S = { v }} .We denote by deg ( v ) = | N ( v ) | the degree of vertex v , as well as δ ( G ) = min v ∈ V ( G ) { deg ( v ) } the minimum degree of G , ∆ ( G ) = max v ∈ V ( G ) { deg ( v ) } the maximum degree of G and n ( G ) = | V ( G ) | the order of G . Given a set S ⊆ V ( G ) , N ( S ) = ∪ v ∈ S N ( v ) , N [ S ] = N ( S ) ∪ S and thesubgraph of G induced by S will be denoted by G [ S ] .A set S ⊆ V ( G ) is a total dominating set of a graph G without isolated vertices if everyvertex v ∈ V ( G ) is adjacent to at least one vertex in S . Let D t ( G ) be the set of total dominatingsets of G . The total domination number of G is defined to be, γ t ( G ) = min {| S | : S ∈ D t ( G ) } . By definition, D t ( G ) ⊆ D ( G ) , so that γ ( G ) ≤ γ t ( G ) . Furthermore, γ t ( G ) ≤ γ ( G ) . Wedefine a γ t ( G ) -set as a set S ∈ D t ( G ) with | S | = γ t ( G ) . The same agreement will be assumedfor optimal parameters associated to other characteristic sets defined in the paper. For instance,a γ ( G ) -set will be a set S ∈ D ( G ) with | S | = γ ( G ) .A graph invariant closely related to the domination number is the packing number. A set S ⊆ V ( G ) is a packing if N [ u ] ∩ N [ v ] = ∅ for every pair of different vertices u , v ∈ S . We define ℘ ( G ) = { S ⊆ V ( G ) : S is a packing of G } . packing number , denoted by ρ ( G ) , is the maximum cardinality among all packings of G ,i.e., ρ ( G ) = max {| S | : S ∈ ℘ ( G ) } . Obviously, γ ( G ) ≥ ρ ( G ) . Furthermore, Meir and Moon [27] showed in 1975 that γ ( T ) = ρ ( T ) for every tree T . We would point out that, in general, these γ ( T ) -sets and ρ ( T ) -sets are notidentical. Notice that D ( G ) ∩ ℘ ( G ) = ∅ if and only if there exists a γ ( G ) -set which is a ρ ( G ) -set. A graph G is an efficient closed domination graph if D ( G ) ∩ ℘ ( G ) = ∅ .A set S ⊆ V ( G ) is an open packing , if N ( u ) ∩ N ( v ) = ∅ for every pair of different vertices u , v ∈ S . We define ℘ o ( G ) = { S ⊆ V ( G ) : S is an open packing of G } . The open packing number of G , denoted by ρ o ( G ) , is the maximum cardinality among all openpackings of G , i.e., ρ o ( G ) = max {| S | : S ∈ ℘ o ( G ) } . By definition, ℘ ( G ) ⊆ ℘ o ( G ) , so that ρ ( G ) ≤ ρ o ( G ) for every graph G , and ρ o ( G ) ≤ γ t ( G ) forevery graph G without isolated vertices. Besides, if S ∈ ℘ o ( G ) , then every vertex of G [ S ] hasdegree at most one, which implies that we can write S = S ∪ S , where S is the set of isolatedvertices of G [ S ] and S = S \ S . Obviously, S = ∅ if and only if S ∈ ℘ ( G ) .A graph G is an efficient open domination graph if there exists a set D , called an efficientopen dominating set , for which V ( G ) = ∪ u ∈ D N ( u ) and N ( u ) ∩ N ( v ) = ∅ for every pair ofdistinct vertices u , v ∈ D . As shown in [23], if G is an efficient open domination graph with anefficient open dominating set D , then γ t ( G ) = | D | . Hence, the following remark holds. Remark 2.1.
A graph G is an efficient open domination graph if and only if there existsS ∈ D p ( G ) such that G [ S ] ∼ = ∪ K . In such a case, | S | = γ t ( G ) = ρ o ( G ) . Corollary 2.2.
If G is an efficient open domination graph, then γ p ( G ) ≤ γ t ( G ) . Given two nontrivial graphs G and H , we define the following properties, which willbecome important tools in the next sections. P ( G , H ) : δ ( H ) = G is an efficient open domination graph. P ( G , H ) : γ ( H ) = G is an efficient closed domination graph. P ( G , H ) : δ ( H ) = G is an efficient open domination graph and γ p ( G ) = γ t ( G ) .Let f : V ( G ) −→ { , , } be a function on G and let V i = { v ∈ V ( G ) : f ( v ) = i } , where i ∈ { , , } . We will identify f with the subsets V , V , V , and so we will use the unifiednotation f ( V , V , V ) for the function and these associated subsets.An RDF f ( V , V , V ) on G is a total Roman dominating function if V ∪ V ∈ D t ( G ) [1].The total Roman domination number , denoted by γ tR ( G ) , is the minimum weight among alltotal Roman dominating functions on G . By definition, γ R ( G ) ≤ γ tR ( G ) .The lexicographic product of two graphs G and H is the graph G ◦ H whose vertex set is V ( G ◦ H ) = V ( G ) × V ( H ) and ( u , v )( x , y ) ∈ E ( G ◦ H ) if and only if ux ∈ E ( G ) or u = x and4 y ∈ E ( H ) . For simplicity, the neighbourhood of ( x , y ) ∈ V ( G ◦ H ) will be denoted by N ( x , y ) instead of N (( x , y )) , and for any PRDF f on G ◦ H we will write f ( x , y ) instead of f (( x , y )) .Notice that for any u ∈ V ( G ) the subgraph of G ◦ H induced by { u } × V ( H ) is isomorphicto H . We will denote this subgraph by H u . For any u ∈ V ( G ) and any function f on G ◦ H wedefine f ( H u ) = ∑ v ∈ V ( H ) f ( u , v ) and f [ H u ] = ∑ x ∈ N [ u ] f ( H x ) . 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 orsharp bounds for specific parameters of the product of two graphs and express them in termsof invariants of the factor graphs. In particular, we cite the following works on dominationtheory of lexicographic product graphs. For instance, the reader is referred to [25, 28] for thedomination number, [4] for the double domination number, [30] for the Roman dominationnumber, [6, 8] for the total Roman domination number, [31] for the rainbow domination num-ber, [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] forthe Italian domination number and [2] for the doubly connected domination number.For the remainder of the paper, definitions will be introduced whenever a concept isneeded.
The next theorem merges two results obtained in [30] and [34].
Theorem 3.1 ([30] and [34]) . For any graph G with no isolated vertex and any nontrivialgraph H, γ ( G ◦ H ) = ( γ ( G ) if γ ( H ) = , γ t ( G ) if γ ( H ) ≥ . As the following result shows, when computing the perfect domination number of lexico-graphic product graphs G ◦ H , where G is connected and H is not trivial, we have to take intoaccount that the class of graphs G ◦ H satisfies a certain trichotomy, as it is divided into threecategories, i.e., the class of graphs G ◦ H for which P ( G , H ) holds, the class of graphs G ◦ H for which P ( G , H ) holds, and the class where nor P ( G , H ) neither P ( G , H ) holds. Theorem 3.2.
For any connected graph G and any nontrivial graph H, γ p ( G ◦ H ) = γ t ( G ) if P ( G , H ) holds, γ ( G ) if P ( G , H ) holds, n ( G ) n ( H ) otherwise. roof. Let S be a γ p ( G ◦ H ) -set and define W = { x ∈ V ( G ) : V ( H x ) ∩ S = ∅ } and W = { x ∈ V ( G ) : | V ( H x ) ∩ S | = } . We differentiate, the following two cases.Case 1. There exists x ∈ V ( G ) such that | V ( H x ) ∩ S | ≥
2. Since n ( H ) ≥
2, we deduce that N [ x ] × V ( H ) ⊆ S , which implies that S = V ( G ◦ H ) , i.e., γ p ( G ◦ H ) = | S | = n ( G ) n ( H ) .Case 2. | V ( H x ) ∩ S | ≤ x ∈ V ( G ) . Obviously, W ∈ D p ( G ) and, since V ( H x ) \ S = ∅ for every x ∈ V ( G ) , we conclude that S ∈ ℘ o ( G ◦ H ) . Let ( x , y ) ∈ S . If x is an isolated vertex of G [ W ] , then y is a universal vertex of H , while if x has degree one, then y is an isolated vertexof H . Therefore, we have the following two complementary subcases.Subcase 2.1. P ( G , H ) holds, i.e., y is an isolated vertex of H , W ∈ D p ( G ) and G [ W ] ∼ = ∪ K .In this case, Remark 2.1 leads to | W | = γ t ( G ) . Hence, γ p ( G ◦ H ) = | S | = | W × { y }| = | W | = γ t ( G ) .Subcase 2.2. P ( G , H ) holds, i.e., y is a universal vertex of H , W is ρ ( G ) -set and also a γ ( G ) -set. In this case, γ p ( G ◦ H ) = | S | = | W × { y }| = | W | = γ ( G ) .The Roman domination number of the lexicographic product of two connected graphs G and H was studied in [30]. Obviously, the connectivity of G ◦ H only depends on theconnectivity of G . Since we need to consider the case where H is not necessarily connected,we make next the necessary modifications to adapt the results obtained in [30] to the generalcase in which H is not necessarily connected. Lemma 3.3.
Let G be a graph with no isolated vertex and H a nontrivial graph. Let f ( V , V , V ) be a γ R ( G ◦ H ) -function, A f = { x ∈ V ( G ) : V ( H x ) ∩ V = ∅ } and B f = { x ∈ V ( G ) \ A f : V ( H x ) ∩ V = ∅ } . If | V | is maximum among all γ R ( G ◦ H ) -functions, then A f ∈ D ( G ) andB f = ∅ .Proof. Let f ( V , V , V ) be a γ R ( G ◦ H ) -function such that | V | is maximum among all γ R ( G ◦ H ) -functions. If x ∈ V ( G ) \ ( A f ∪ B f ) , then V ( H x ) ⊆ V , which implies that N ( x ) ∩ A f = ∅ .Hence, A f ∪ B f ∈ D ( G ) .Now, suppose that there exists u ∈ B f . Observe that ( N ( u ) × V ( H )) ∩ V = ∅ , and so V ( H u ) ⊆ V . Given u ′ ∈ N ( u ) and v ∈ V ( H ) , we define a function f ′ ( V ′ , V ′ , V ′ ) on G ◦ H by f ′ ( H u ) = f ′ ( u ′ , v ) = f ′ ( x , y ) = f ( x , y ) for the remaining vertices. Notice that f ′ is aRDF on G ◦ H with | V ′ | > | V | and, since H is a nontrivial graph, f ( H u ) = | V ( H u ) | ≥
2, so that ω ( f ′ ) ≤ ω ( f ) , which is a contradiction. Therefore, B f = ∅ and A f ∈ D ( G ) .The following result is a direct consequence of Lemma 3.3. Corollary 3.4.
For any graph G without isolated vertices and any nontrivial graph H, γ R ( G ◦ H ) ≥ γ ( G ) . Theorem 3.5. [30]
For any graph G without isolated vertices and any graph H, γ R ( G ◦ H ) ≤ γ t ( G ) . ( A , B ) of disjoint sets A , B ⊆ V ( G ) is a dominating couple of G if every vertex x ∈ V ( G ) \ B satisfies that N ( x ) ∩ ( A ∪ B ) = ∅ . Also, we define the parameter ζ ( G ) as follows. ζ ( G ) = min { | A | + | B | : ( A , B ) is a dominating couple of G } . We say that a dominating couple ( A , B ) of G is a ζ ( G ) -couple if ζ ( G ) = | A | + | B | . With thisnotation in mind, we state the following result. Theorem 3.6.
For any graph G without isolated vertices and any nontrivial graph H, γ R ( G ◦ H ) = γ ( G ) if ∆ ( H ) = n ( H ) − , ζ ( G ) if ∆ ( H ) = n ( H ) − , γ t ( G ) if ∆ ( H ) ≤ n ( H ) − .Proof. As shown in [30], if γ ( H ) = G is a connected nontrivial graph, then γ R ( G ◦ H ) = γ ( G ) . Obviously, the same equality holds if G is not connected.In order to discuss the remaining cases, let f ( V , V , V ) be a γ R ( G ◦ H ) -function such that | V | is maximum. By Lemma 3.3, A f = { x ∈ V ( G ) : V ( H x ) ∩ V = ∅ } is a dominating set of G and B f = { x ∈ V ( G ) \ A f : V ( H x ) ∩ V = ∅ } = ∅ . Let A ′ f = { x ∈ A f : N ( x ) ∩ A f = ∅ } .Assume ∆ ( H ) = n ( H ) −
2. Since ( A f \ A ′ f , A ′ f ) is a dominating couple of G , we deduce that ζ ( G ) ≤ | A f \ A ′ f | + | A ′ f | = ω ( f ) = γ R ( G ◦ H ) . Now, let v ∈ V ( H ) be a vertex of maximumdegree and { v ′ } = V ( H ) \ N [ v ] . Since for any ζ ( G ) -couple ( A , B ) , the function g ( W , W , W ) ,defined by W = ( A ∪ B ) × { v } and W = B × { v ′ } , is an RDF on G ◦ H , we deduce that γ R ( G ◦ H ) ≤ ω ( g ) = | W | + | W | = | A | + | B | = ζ ( G ) . Therefore, γ R ( G ◦ H ) = ζ ( G ) .Finally, assume ∆ ( H ) ≤ n ( H ) −
3. By Theorem 3.5 we only need to prove that γ R ( G ◦ H ) ≥ γ t ( G ) . In this case, if x ∈ A ′ f , then f ( H x ) ≥
4, while if x ∈ A f \ A ′ f , then f ( H x ) ≥
2. Since G does not have isolated vertices and A f ∈ D ( G ) , we have that γ t ( G ) ≤ | A f \ A ′ f | + | A ′ f | . Hence,2 γ t ( G ) ≤ | A f \ A ′ f | + | A ′ f | ≤ ω ( f ) = γ R ( G ◦ H ) , which completes the proof.Two simple characterizations of Roman graphs were given in [10], but the authors sug-gest finding classes of Roman graphs. The following result is an immediate consequence ofTheorems 3.1 and 3.6. Theorem 3.7.
Let G be a graph with no isolated vertex. If H is a graph such that ∆ ( H ) = n ( H ) − , then G ◦ H is a Roman graph. As ζ ( G ) has not been extensively studied, we next obtain tight bounds on γ R ( G ◦ H ) forthe case in which ∆ ( H ) = n ( H ) − Theorem 3.8.
Let G a graph with no isolated vertex and H a graph. If ∆ ( H ) = n ( H ) − , then max { γ tR ( G ) , γ t ( G ) + γ ( G ) } ≤ γ R ( G ◦ H ) ≤ min { γ ( G ) , γ t ( G ) } . Proof.
Let f ( V , V , V ) be a γ R ( G ◦ H ) -function with | V | maximum. As above, let A f = { x ∈ V ( G ) : V ( H x ) ∩ V = ∅ } , B f = { x ∈ V ( G ) \ A f : V ( H x ) ∩ V = ∅ } and A ′ f = { x ∈ A f :7 ( x ) ∩ A f = ∅ } . By Lemma 3.3, B f = ∅ and A f ∈ D ( G ) . Furthermore, if x ∈ A ′ f , then f ( H x ) =
3, while if x ∈ A f \ A ′ f , then f ( H x ) =
2. Thus, γ R ( G ◦ H ) = | A ′ f | + | A f \ A ′ f | . We first prove the lower bounds. Let S ⊆ V ( G ) be a set of minimum cardinality among thesets satisfying that A f ⊆ S and S ∩ N ( x ) = ∅ for every vertex x ∈ A ′ f . Since S ∈ D t ( G ) , wededuce that γ t ( G ) ≤ | S | ≤ | A ′ f | + | A f \ A ′ f | . Hence, γ t ( G ) + γ ( G ) ≤ ( | A ′ f | + | A f \ A ′ f | ) + | A f | = | A ′ f | + | A f \ A ′ f | = γ R ( G ◦ H ) .Now, let g ( W , W , W ) be a function on G defined by W = A f and W = S \ A f . Notice that g is a TRDF on G . Thus, γ tR ( G ) ≤ ω ( g ) = | A f | + | S \ A f | ≤ | A ′ f | + | A f \ A ′ f | = γ R ( G ◦ H ) ,which completes the proof of the lower bounds.In order to prove the upper bounds, let D be a γ ( G ) -set, and let v , v ′ ∈ V ( H ) such that v isa vertex of maximum degree and { v ′ } = V ( H ) \ N [ v ] . Notice that the function f ′ ( V ′ , V ′ , V ′ ) ,defined by V ′ = D × { v } and V ′ = D × { v ′ } , is an RDF on G ◦ H . Therefore, γ R ( G ◦ H ) ≤ ω ( f ′ ) = | D | = γ ( G ) .Finally, the bound γ R ( G ◦ H ) ≤ γ t ( G ) is already known from Theorem 3.5. Therefore,the proof is complete.The bounds above are tight. Notice that, if γ t ( G ) = γ ( G ) , then γ R ( G ◦ H ) = γ tR ( G ) = γ t ( G ) , while if γ t ( G ) = γ ( G ) , then we have γ R ( G ◦ H ) = γ t ( G ) + γ ( G ) = γ ( G ) . This section is organised as follows. First we obtain tight bounds on γ pR ( G ◦ H ) and then wegive sufficient and/or necessary conditions for the bounds to be achieved. We also discuss thecase of perfect Roman graphs and we characterize the graphs where γ pR ( G ◦ H ) = γ R ( G ◦ H ) . Theorem 4.1.
For any graph G without isolated vertices and any graph H, γ pR ( G ◦ H ) ≤ γ p ( G )( n ( H ) + ) . Proof.
Let S be a γ p ( G ) -set and v ∈ V ( H ) . Let f ( V , V , V ) be a function on G ◦ H defined by V = S × { v } and V = S × ( V ( H ) \ { v } ) . Clearly, f is a PRDF, which implies that γ pR ( G ◦ H ) ≤ ω ( f ) = | S | + | S | ( n ( H ) − ) = γ p ( G )( n ( H ) + ) . Therefore, the result follows.In order to see that the bound above is tight, we can consider the corona graph G ∼ = G ′ ⊙ N k ,where k ≥ G ′ is any graph of minimum degree at least two, and H is a nontrivial graph. Inthis case, γ pR ( G ◦ H ) = n ( G ′ )( n ( H ) + ) = γ p ( G )( n ( H ) + ) . Theorem 4.2.
Let G be a graph without isolated vertices and H a graph. The followingstatements hold. (i)
For any γ pR ( G ) -function f ( V , V , V ) , γ pR ( G ◦ H ) ≤ γ pR ( G ) + ( | V | + | V | )( n ( H ) − ) . If there exists a γ pR ( G ) -function f ( V , V , V ) such that V is a γ ( G ) -set, then γ pR ( G ◦ H ) ≤ γ pR ( G ) n ( H ) − γ ( G )( n ( H ) − ) . (iii) If S is a γ p ( G ) -set, S ′ = { x ∈ S : epn ( x , S ) = ∅ } and S ′′ = S \ S ′ , then γ pR ( G ◦ H ) ≤ | S ′ | + | S ′′ | + γ p ( G )( n ( H ) − ) . (iv) If there exists a γ pR ( G ) -function f ( V , V , V ) such that V ∪ V is a γ p ( G ) -set, then γ pR ( G ◦ H ) ≤ γ pR ( G ) + γ p ( G )( n ( H ) − ) . Proof.
From any γ pR ( G ) -function f ( V , V , V ) , we can define a function g ( W , W , W ) on G ◦ H as W = V × { v } and W = V × ( V ( H ) \ { v } ) ∪ V × V ( H ) . It is readily seen that g is a PRDFand, as a result, γ pR ( G ◦ H ) ≤ ω ( g ) = | V | + | V | ( n ( H ) − ) + | V | n ( H ) = γ pR ( G ) + ( | V | + | V | )( n ( H ) − ) . Therefore, (i) follows.Now, since γ pR ( G ) + ( | V | + | V | )( n ( H ) − ) = γ pR ( G ) n ( H ) − | V | ( n ( H ) − ) , from (i) wededuce (ii).In order to prove (iii), we only need to observe that for any γ p ( G ) -set S , the function h ( V ( G ) \ S , S ′ , S ′′ ) is a PRDF on G . Thus, we conclude the proof of (iii) by analogy to theproof of (i), by using h instead of f .Finally, (iv) follows from (i).The bounds above are tight. For instance, let G be the graph shown in Figure 2, V = S ′′ the set of vertices labelled with 2, V = S ′ the set of vertices labelled with 1 and V = V ( G ) \ ( V ∪ V ) . In this case, V is a γ ( G ) -set, f ( V , V , V ) is a γ pR ( G ) -function, S = S ′ ∪ S ′′ is a γ p ( G ) -set and γ pR ( G ◦ H ) = ( H ) + H . Therefore, the bounds above areachieved. Theorem 4.3.
For any graph G without isolated vertices and any graph H, γ pR ( G ◦ H ) ≤ min S ∈ ℘ o ( G ) {| S | ( n ( H ) − ∆ ( H ) + ) + | S | ( + δ ( H )) + n ( H )( n ( G ) − | N [ S ] | ) } . Proof.
Let S = S ∪ S ∈ ℘ o ( G ) and y , y ∈ V ( H ) such that deg ( y ) = δ ( H ) and deg ( y ) = ∆ ( H ) . From S , y and y , we can construct a function f ( V , V , V ) on G ◦ H as follows. Let V = S × { y } ∪ S × { y } and V = S × ( V ( H ) \ N [ y ]) ∪ S × N ( y ) ∪ ( V ( G ) \ N [ S ]) × V ( H ) .It is readily seen that f is a PRDF on G ◦ H . Therefore, γ pR ( G ◦ H ) ≤ ω ( f ) = | S | ( n ( H ) − ∆ ( H ) + ) + | S | ( + δ ( H )) + n ( H )( n ( G ) − | N [ S ] | ) . Since the inequality holds for any openpacking of G , the result follows.The following result is an immediate consequence of Theorem 4.3. Corollary 4.4.
Given a graph G without isolated vertices, the following statements hold. (i)
If G is an efficient open domination graph, then for any graph H, γ pR ( G ◦ H ) ≤ γ t ( G )( + δ ( H )) . If G is an efficient closed domination graph, then for any graph H, γ pR ( G ◦ H ) ≤ γ ( G )( n ( H ) − ∆ ( H ) + ) . Proof.
First, we proceed to prove (i). Let S ∈ D p ( G ) such that G [ S ] ∼ = ∪ K . Notice that S = S ∈ ℘ o ( G ) and N [ S ] = V ( G ) . Hence, by Theorem 4.3 and Remark 2.1 we deduce that γ pR ( G ◦ H ) ≤ | S | ( + δ ( H )) = γ t ( G )( + δ ( H )) . Finally, we proceed to prove (ii). Let S be a γ ( G ) -set which is a ρ ( G ) -set. Since S = S ∈ ℘ o ( G ) and N [ S ] = V ( G ) , by Theorem 4.3 we deduce that γ pR ( G ◦ H ) ≤ | S | ( n ( H ) − ∆ ( H ) + ) = ρ ( G )( n ( H ) − ∆ ( H ) + ) . As we will show in Theorems 4.5 and 4.8, the bounds above are tight.
Theorem 4.5.
Given a nontrivial graph G with γ ( G ) = , the following statements hold. (i) If δ ( G ) ≥ , then for any graph H, γ pR ( G ◦ H ) = n ( H ) − ∆ ( H ) + . (ii) If δ ( G ) = , then for any graph H, γ pR ( G ◦ H ) = min { δ ( H ) + , n ( H ) − ∆ ( H ) + } . Proof.
Let f ( V , V , V ) be a γ pR ( G ◦ H ) -function. We assume first that δ ( G ) ≥
2. Notice that,in such a case, N ( x ) ∩ N ( x ′ ) = ∅ for any x , x ′ ∈ V ( G ) . We differentiate three cases for V .Case 1. There exists x ∈ V ( G ) such that | V ∩ V ( H x ) | ≥
2. In this case, f ( H x ′ ) = n ( H ) forevery x ′ ∈ N ( x ) , and so γ pR ( G ◦ H ) = ω ( f ) ≥ f [ H x ] ≥ + n ( H ) , which is a contradiction withCorollary 4.4-(ii).Case 2: There exist two different vertices ( x , y ) , ( x ′ , y ′ ) ∈ V such that x = x ′ . In this case, f ( H z ) = n ( H ) for any z ∈ N ( x ) ∩ N ( x ′ ) , and so γ pR ( G ◦ H ) = ω ( f ) ≥ f [ H z ] ≥ + n ( H ) , whichis again a contradiction with Corollary 4.4-(ii).Case 3: V = { ( x , y ) } . In this case, f ( x , v ) = v ∈ V ( H ) \ N [ y ] . Hence, γ pR ( G ◦ H ) = ω ( f ) ≥ f ( H x ) ≥ n ( H ) − deg ( y ) + ≥ n ( H ) − ∆ ( H ) +
1. By Corollary 4.4-(ii) we concludethat γ pR ( G ◦ H ) = n ( H ) − ∆ ( H ) + . According to the three cases above, (i) follows.From now on we assume that δ ( G ) = V .Case 1’: There exists x ∈ V ( G ) such that | V ∩ V ( H x ) | ≥
2. As in Case 1, we obtain a contra-diction.Case 2’: There exist two different vertices ( x , y ) , ( x ′ , y ′ ) ∈ V such that x = x ′ . If deg ( x ) < ∆ ( G ) − ( x ′ ) < ∆ ( G ) −
1, then f ( H z ) = n ( H ) for every z ∈ N ( x ) ∩ N ( x ′ ) , and so γ pR ( G ◦ H ) = ω ( f ) ≥ f [ H z ] ≥ + n ( H ) , which is a contradiction with Corollary 4.4-(ii).Now, assume that deg ( x ) = ∆ ( G ) −
1. If deg ( x ′ ) ≥
2, then as above f ( H z ) = n ( H ) forevery z ∈ N ( x ) ∩ N ( x ′ ) , and we have again a contradiction with Corollary 4.4-(ii). Finally, ifdeg ( x ′ ) =
1, then f ( x , b ) ≥ b ∈ N ( y ) and f ( x ′ , b ′ ) ≥ b ′ ∈ N ( y ′ ) . Thus,10 pR ( G ◦ H ) = ω ( f ) ≥ f ( H x ) + f ( H x ′ ) ≥ δ ( H ) +
4, and by Corollary 4.4-(i) we conclude that γ pR ( G ◦ H ) = δ ( H ) + V = { ( x , y ) } . As in Case 3, we deduce that γ pR ( G ◦ H ) = n ( H ) − ∆ ( H ) + . According to these last three cases, (ii) follows.
Lemma 4.6.
Let f ( V , V , V ) be a γ pR ( G ◦ H ) -function and x ∈ V ( G ) . If V ( H x ) ∩ V = ∅ , theneither V ( H x ) ⊆ V or V ( H x ) ⊆ V .Proof. Suppose that V ( H x ) ∩ V = ∅ and there exist y , y ∈ V ( H ) such that f ( x , y ) = f ( x , y ) =
1. In such a case, there exists exactly one vertex ( u , v ) ∈ V which is adjacent to ( x , y ) . Hence, u ∈ N ( x ) and ( u , v ) is the only vertex belonging to V which is adjacent to ( x , y ) . Thus, the function g ( W , W , W ) , defined by W = V , W = V \ V ( H x ) and W = V ∪ V ( H x ) , is a PRDF on G ◦ H , which is a contradiction, as ω ( g ) < ω ( f ) . Therefore, theresult follows. Theorem 4.7.
For any graph G without isolated vertices and any nontrivial graph H, γ pR ( G ◦ H ) ≥ γ ( G ) min { n ( H ) − ∆ ( H ) + , + δ ( H ) } . Proof.
Let f ( V , V , V ) be a γ pR ( G ◦ H ) -function, and define W = { x ∈ V ( G ) : V ( H x ) ⊆ V } , W = { x ∈ V ( G ) : V ( H x ) ⊆ V } and W = V ( G ) \ ( W ∪ W ) . In fact, by Lemma 4.6, W = { x ∈ V ( G ) : V ( H x ) ∩ V = /0 } . Let W , be the set of isolated vertices of G [ W ] , W , = W \ W , and W , = { x ∈ W , : N ( x ) × V ( H ) ∩ V = ∅ } .Thus, if x ∈ W , then V ( H x ) ⊆ V and there exists exactly one vertex ( u , v ) ∈ V such that u ∈ N ( x ) ∩ W . Also, if x ∈ W , \ W , , then N ( x ) ∩ W = ∅ . Hence, W ∪ W , ∪ W , ∈ D ( G ) .Notice that if x ∈ W , , then f ( H x ) ≥ n ( H ) − ∆ ( H ) + x ∈ W , , then f ( H x ) ≥ + δ ( H ) .Therefore, γ pR ( G ◦ H ) = ∑ x ∈ V ( G ) f ( H x ) ≥ ∑ x ∈ W , f ( H x ) + ∑ x ∈ W , f ( H x ) + ∑ x ∈ W f ( H x ) ≥ | W , | ( n ( H ) − ∆ ( H ) + ) + | W , | ( + δ ( H )) + | W | n ( H ) ≥ ( | W , | + | W , | + | W | ) min { n ( H ) − ∆ ( H ) + , + δ ( H ) }≥ γ ( G ) min { n ( H ) − ∆ ( H ) + , + δ ( H ) } . From Corollary 4.4 and Theorem 4.7 we deduce the following result.
Theorem 4.8.
Given a graph G without isolated vertices, the following statements hold. (i)
If G is an efficient closed domination graph, then for any graph H with ≤ n ( H ) ≤ ∆ ( H ) + δ ( H ) + , γ pR ( G ◦ H ) = γ ( G )( n ( H ) − ∆ ( H ) + ) . If γ p ( G ) = γ t ( G ) = γ ( G ) and G is an efficient open domination graph, then for anynontrivial graph H with n ( H ) ≥ ∆ ( H ) + δ ( H ) + , γ pR ( G ◦ H ) = γ ( G )( + δ ( H )) . Corollary 4.9.
Given a graph G without isolated vertices and a nontrivial graph H, the fol-lowing statements hold. (i) If P ( G , H ) holds, then γ pR ( G ◦ H ) = γ ( G ) . (ii) If γ p ( G ) = γ ( G ) and P ( G , H ) holds, then γ pR ( G ◦ H ) = γ ( G ) . Theorem 4.10.
Given two nontrivial graphs G and H, the following statements hold. (i) γ pR ( G ◦ H ) ≥ max { γ pR ( G ) , γ ( G ) } . (ii) γ pR ( G ◦ H ) = γ pR ( G ) if and only if γ pR ( G ) = γ p ( G ) and either P ( G , H ) holds or P ( G , H ) holds. (iii) If H has order at least three, then γ pR ( G ◦ H ) = γ ( G ) if and only if γ p ( G ) = γ ( G ) andeither P ( G , H ) holds or P ( G , H ) holds.Proof. By Theorem 4.7 we deduce that γ pR ( G ◦ H ) ≥ γ ( G ) . From now on, let f ( V , V , V ) be a γ pR ( G ◦ H ) -function, and define the function g ( W , W , W ) on G by W = { x ∈ V ( G ) : V ( H x ) ⊆ V } , W = { x ∈ V ( G ) : V ( H x ) ⊆ V } and W = V ( G ) \ ( W ∪ W ) . If x ∈ W , then V ( H x ) ⊆ V and there exists exactly one vertex ( u , v ) ∈ V such that u ∈ N ( x ) ∩ W . Hence, g is a PRDF on G , and so γ pR ( G ) ≤ ω ( g ) ≤ ω ( f ) = γ pR ( G ◦ H ) . Therefore, (i) follows.In order to prove (ii), assume γ pR ( G ◦ H ) = γ pR ( G ) . Notice that in this case the function g ( W , W , W ) defined above is a γ pR ( G ) -function. We first show that the γ pR ( G ◦ H ) -function f ( V , V , V ) satisfies V = ∅ . Suppose to the contrary, that there exists ( u , v ) ∈ V . If V ( H u ) ∩ V = ∅ , then by Lemma 4.6 we have that V ( H u ) ⊆ V and since | V ( H u ) | ≥
2, we deducethat γ pR ( G ) ≤ ω ( g ) < ω ( f ) = γ pR ( G ◦ H ) , which is a contradiction. The same contradiction isreached if V ( H u ) ∩ V = ∅ , as in such a case f ( H u ) ≥
3. Hence, V = ∅ , which implies that W = ∅ and W ∈ D p ( G ) .Furthermore, 2 γ p ( G ) ≤ | W | ≤ γ pR ( G ◦ H ) = γ pR ( G ) ≤ γ p ( G ) , and so we conclude that W is a γ p ( G ) -set and γ pR ( G ) = γ p ( G ) . We differentiate two cases for x ∈ W .Case 1. There exists x ′ ∈ N ( x ) ∩ W . In this case, there exist y , y ′ ∈ V ( H ) such that ( x , y ) , ( x ′ , y ′ ) ∈ V , and so no vertex in V ( H x ) \ { ( x , y ) } is adjacent to ( x , y ) . Hence y is an isolated vertex of H . Notice that N ( x ) ∩ W = { x ′ } , otherwise every vertex in V ( H x ) ∩ V = V ( H x ) \ { ( x , y ) } isadjacent to two vertices in V , which is a contradiction.Case 2. N ( x ) ∩ W = ∅ . In this case, there exists y ∈ V ( H ) such that ( x , y ) ∈ V and everyvertex in V ( H x ) ∩ V = V ( H x ) \ { ( x , y ) } has to be adjacent to ( x , y ) . Hence, y is a universalvertex of H and so γ ( H ) =
1. Notice also that N ( x ) ∩ N ( x ′ ) = ∅ for every x ′ ∈ W \ { x } . According to the two cases above, either H has at least one isolated vertex or γ ( H ) = x ∈ W or Case 2 holds for every vertex x ∈ W .In the first case, it is readily seen that P ( G , H ) holds, while if Case 2 holds for every vertex12 ∈ W , then W is a packing, and so γ p ( G ) = | W | ≤ ρ ( G ) ≤ γ ( G ) ≤ γ p ( G ) , which impliesthat P ( G , H ) holds.Conversely, assume that γ pR ( G ) = γ p ( G ) . If P ( G , H ) holds, then Corollary 4.4-(i) andthe lower bound (i) lead to γ pR ( G ◦ H ) = γ pR ( G ) . Finally, if P ( G , H ) holds, then Theorem4.8-(i) leads to γ pR ( G ◦ H ) = γ pR ( G ) , which completes the proof of (ii).We proceed to prove (iii). Assume γ pR ( G ◦ H ) = γ ( G ) . Since | V ( H ) | ≥ W ∪ W ∈ D ( G ) , we deduce that if W = ∅ , then 2 γ ( G ) < | V ( H ) || W | + | W | ≤ ω ( f ) = γ pR ( G ◦ H ) , which is a contradiction. Hence, W = ∅ and W ∈ D p ( G ) . Furthermore, 2 γ ( G ) ≤ | W | = γ pR ( G ◦ H ) = γ ( G ) , which implies that W is a γ ( G ) -set and also a γ p ( G ) -set. We differentiatetwo cases for x ∈ W .Case 1’. There exists x ′ ∈ N ( x ) ∩ W . As in Case 1, we can see that H has an isolated vertexand N ( x ) ∩ W = { x ′ } .Case 2’. N ( x ) ∩ W = ∅ . By analogy to Case 2 we deduce that γ ( H ) = x ∈ W or Case 2’ holds for every vertex x ∈ W . In the first case, we deduce that P ( G , H ) follows, while if Case 2’ holds for everyvertex x ∈ W , then W is a packing, and so γ p ( G ) = | W | ≤ ρ ( G ) ≤ γ ( G ) ≤ γ p ( G ) , whichleads to P ( G , H ) .Conversely, assume γ p ( G ) = γ ( G ) . If P ( G , H ) holds, then Corollary 4.4-(i) and the lowerbound (i) lead to γ pR ( G ◦ H ) = γ ( G ) . Finally, if P ( G , H ) holds, then Theorem 4.8-(i) leads to γ pR ( G ◦ H ) = γ ( G ) , which completes the proof. Theorem 4.11.
Let G and H be two graphs. If G is an efficient open domination graph and n ( H ) ≥ ∆ ( H ) + δ ( H ) + , then γ pR ( G ◦ H ) = γ t ( G )( + δ ( H )) . Proof.
Let S ∈ D p ( G ) such that G [ S ] ∼ = ∪ K and assume that n ( H ) ≥ ∆ ( H ) + δ ( H ) + x , x ′ ∈ S be two adjacent vertices, and define X x = { x } ∪ epn ( x , S ) = N [ x ] \ { x ′ } and X x ′ = { x ′ } ∪ epn ( x ′ , S ) = N [ x ′ ] \ { x } . Let f ( V , V , V ) be a γ pR ( G ◦ H ) -function and define ε ( x , x ′ ) = f ( X x × V ( H )) + f ( X x ′ × V ( H )) . In order to prove that ε ( x , x ′ ) ≥ ( + δ ( H )) , wedifferentiate the following cases.Case 1: V ∩ V ( H x ) = V ∩ V ( H x ′ ) = ∅ . By Lemma 4.6 we have that V ( H x ) ⊆ V or V ( H x ) ⊆ V , and also V ( H x ′ ) ⊆ V or V ( H x ′ ) ⊆ V . The case V ( H x ) ⊆ V and V ( H x ′ ) ⊆ V leads to ε ( x , x ′ ) ≥ f ( H x ) + f ( H x ′ ) = | V ( H ) | ≥ ( + δ ( H )) .If V ( H x ) ⊆ V and V ( H x ′ ) ⊆ V , then | V ∩ ( epn ( x , S ) × V ( H )) | =
1, which implies that ε ( x , x ′ ) ≥ f ( X x × V ( H )) + | V ( H x ′ ) | ≥ + | V ( H ) | ≥ + δ ( H ) > ( + δ ( H )) .Finally, if V ( H x ) ⊆ V and V ( H x ′ ) ⊆ V , then | V ∩ ( epn ( x , S ) × V ( H )) | = | V ∩ ( epn ( x ′ , S ) × V ( H )) | =
1. Since n ( H ) ≥ ∆ ( H ) + δ ( H ) +
3, the vertex of weight two inepn ( x , S ) × V ( H ) is not able to dominate every vertex in epn ( x , S ) × V ( H ) , which implies that f ( epn ( x , S ) × V ( H )) ≥ δ ( H ) + f ( epn ( x , S ) × V ( H )) ≥ n ( H ) − ∆ ( H ) + ≥ ( + δ ( H )) .By applying the same reasoning to epn ( x ′ , S ) × V ( H ) we conclude that ε ( x , x ′ ) ≥ f ( epn ( x , S ) × V ( H )) + f ( epn ( x ′ , S ) × V ( H )) ≥ ( + δ ( H )) .Case 2: V ∩ V ( H x ) = ∅ and V ∩ V ( H x ′ ) = ∅ . By Lemma 4.6, either V ( H x ′ ) ⊆ V or V ( H x ′ ) ⊆ V . If V ( H x ′ ) ⊆ V , then ε ( x , x ′ ) ≥ + | V ( H x ′ ) | ≥ + δ ( H ) > ( + δ ( H )) . Now, assume13 ( H x ′ ) ⊆ V . In this case, | V ∩ V ( H x ) | = ( H ) ≥ ∆ ( H ) + δ ( H ) +
3, we havethat ( V ( H x ) \ N [ x ]) ∩ V = /0 or | V ∩ ( epn ( x , S ) × V ( H )) | =
1. In both cases we deduce that ε ( x , x ′ ) ≥ ( + δ ( H )) .Case 3: V ∩ V ( H x ) = ∅ and V ∩ V ( H x ′ ) = ∅ . In this case, | V ∩ V ( H x ) | = | V ∩ V ( H x ) | =
1, which implies that ε ( x , x ′ ) ≥ f ( H x ) + f ( H x ′ ) ≥ ( + δ ( H )) .According to the three cases above we conclude that ε ( x , x ′ ) ≥ ( + δ ( H )) for every pairof adjacent vertices x , x ′ ∈ S . Hence, γ pR ( G ◦ H ) = ω ( f ) ≥ ∑ x ∈ S f ( X x × V ( H )) ≥ | S | ( + δ ( H )) = γ t ( G )( + δ ( H )) . Therefore, Corollary 4.4-(i) leads to γ pR ( G ◦ H ) = γ t ( G )( + δ ( H )) . From the following inequalities we can derive results on the perfect Roman dominationnumber of G ◦ H . γ R ( G ◦ H ) ≤ γ pR ( G ◦ H ) ≤ γ p ( G ◦ H ) . Next we discuss the cases in which the bounds are sharp.
Theorem 4.12.
Given a connected nontrivial graph G and H a nontrivial graph, the followingstatements hold. (i) If ∆ ( H ) = n ( H ) − , then γ pR ( G ◦ H ) = γ p ( G ◦ H ) if and only if P ( G , H ) holds. (ii) If ∆ ( H ) = n ( H ) − , then the following statements hold. (a) If γ pR ( G ◦ H ) = γ p ( G ◦ H ) , then P ( G , H ) holds and γ t ( G ) ≤ | S | + | S | forevery S ∈ ℘ o ( G ) ∩ D ( G ) . (b) If γ pR ( G ) = γ t ( G ) or γ t ( G ) = γ ( G ) , then γ pR ( G ◦ H ) = γ p ( G ◦ H ) if and only if P ( G , H ) holds. (iii) If ∆ ( H ) ≤ n ( H ) − , then γ pR ( G ◦ H ) = γ p ( G ◦ H ) if and only if P ( G , H ) holds.Proof. Assume γ pR ( G ◦ H ) = γ p ( G ◦ H ) . Since G is a graph without isolated vertices and H a nontrivial graph, γ pR ( G ◦ H ) < n ( G ) n ( H ) , so that from Theorem 3.2 we have that either P ( G , H ) holds or P ( G , H ) holds. Notice that, by definition, P ( G , H ) is associated with ∆ ( H ) = n ( H ) − P ( G , H ) holds, then Theorem 3.2 leads to γ pR ( G ◦ H ) ≤ γ p ( G ◦ H ) = γ ( G ) . Insuch a case, from Theorem 4.10-(i) we conclude that γ pR ( G ◦ H ) = γ p ( G ◦ H ) . Therefore, (i)follows.From now on, assume that P ( G , H ) holds. Notice that, in this case, Theorem 3.2 leads to γ pR ( G ◦ H ) ≤ γ p ( G ◦ H ) = γ t ( G ) . (1)First, consider the case ∆ ( H ) = n ( H ) −
2. Let v , v ′ ∈ V ( H ) such that deg ( v ) = n ( H ) − ( v ′ ) =
0. Now, if there exists S ∈ ℘ o ( G ) ∩ D ( G ) such that 2 γ t ( G ) > | S | + | S | thenthe function g ( X , X , X ) , defined by X = S × { v ′ } ∪ S × { v } and X = S × { v ′ } , is a PRDF14n G ◦ H , and so γ pR ( G ◦ H ) ≤ ω ( g ) = | S | + | S | < γ t ( G ) = γ p ( G ◦ H ) . Therefore, (ii)-(a)follows.Furthermore, if γ pR ( G ) = γ t ( G ) , then by Theorem 4.10-(i), 2 γ t ( G ) = γ pR ( G ) ≤ γ pR ( G ◦ H ) and so Eq. (1) implies that γ pR ( G ◦ H ) = γ p ( G ◦ H ) . The case γ t ( G ) = γ ( G ) is analogous tothe previous one. Therefore, (ii)-(b) follows.Finally, if ∆ ( H ) ≤ n ( H ) −
3, then by Theorem 4.11 we have that γ pR ( G ◦ H ) = γ t ( G ) .Hence, Eq. (1) implies that γ pR ( G ◦ H ) = γ p ( G ◦ H ) , which completes the proof of (iii).In order to state the next result, we define the following parameter. ζ ′ ( G ) = min S ∈ ℘ o ( G ) ∩ D ( G ) { | S | + | S |} . A set S ∈ ℘ o ( G ) ∩ D ( G ) of cardinality | S | = ζ ′ ( G ) will be called a ζ ′ ( G ) -set.The following straightforward lemma will be used in the proof of our next result. Lemma 4.13.
A graph G is a perfect Roman graph if and only if there exists a γ pR ( G ) -functionf ( V , V , V ) such that V = ∅ . Theorem 4.14.
The following statements hold for a connected nontrivial graph G and anygraph H of order at least three. (i) If ∆ ( H ) = n ( H ) − , then γ pR ( G ◦ H ) = γ R ( G ◦ H ) if and only if P ( G , H ) holds. (ii) If ∆ ( H ) = n ( H ) − , then γ pR ( G ◦ H ) = γ R ( G ◦ H ) if and only if there exists a ζ ( G ) -couple ( A , B ) such that A ∪ B ∈ ℘ o ( G ) and A = ∅ whenever δ ( H ) ≥ . (iii) If ∆ ( H ) = n ( H ) − , then γ pR ( G ◦ H ) = γ R ( G ◦ H ) if and only if either δ ( H ) = and ζ ′ ( G ) = γ t ( G ) or δ ( H ) ≥ and γ t ( G ) = γ p ( G ) = ρ ( G ) . (iv) If ∆ ( H ) ≤ n ( H ) − , then γ pR ( G ◦ H ) = γ R ( G ◦ H ) if and only if P ( G , H ) holds.Proof. First, assume γ pR ( G ◦ H ) = γ R ( G ◦ H ) . Let f ( V , V , V ) be a γ pR ( G ◦ H ) -function, anddefine W = { x ∈ V ( G ) : V ( H x ) ⊆ V } , W = { x ∈ V ( G ) : V ( H x ) ⊆ V } and W = V ( G ) \ ( W ∪ W ) . Notice that f is also a γ R ( G ◦ H ) -function. If there exists u ∈ W , then for any u ′ ∈ N ( u ) and v ∈ V ( H ) , the function g , defined by g ( H u ) = g ( u ′ , v ) =
2, and g ( x , y ) = f ( x , y ) for theremaining vertices, is an RDF on G ◦ H of weight ω ( g ) < ω ( f ) = γ pR ( G ◦ H ) = γ R ( G ◦ H ) ,which is a contradiction. Hence, W = ∅ . Now, suppose that G [ W ] has a vertex x of degreeat least two. Since f is a γ pR ( G ◦ H ) -function, V ( H x ) ∩ V = ∅ and, since f is a γ R ( G ◦ H ) -function, V ( H x ) ∩ V = ∅ , which is a contradiction. Therefore, W ∈ ℘ o ( G ) ∩ D p ( G ) . Wedifferentiate two cases. From each case, we will get partial conclusions and, once both caseshave been analysed, we will be able to complete the proof of each statement separately.Case 1. V = ∅ . Lemma 4.13 leads to γ pR ( G ◦ H ) = γ p ( G ◦ H ) . Hence, if ∆ ( H ) = n ( H ) − P ( G , H ) holds. Analogously, if ∆ ( H ) ≤ n ( H ) − P ( G , H ) holds. Notice that in this latter case δ ( H ) = γ R ( G ◦ H ) = γ pR ( G ◦ H ) = γ p ( G ◦ H ) = γ t ( G ) .Case 2. V = ∅ . Let u ∈ V ( G ) such that V ( H u ) ∩ V = ∅ . Since W = ∅ , by Lemma 4.6 wehave that u ∈ W . Since f is also a γ R ( G ◦ H ) -function, N ( u ) ∩ W = ∅ and so N ( u ) ⊆ W , which15mplies that W ′ = { x ∈ W : V ( H x ) ∩ V = ∅ } is a packing. Notice also that | V ( H u ) ∩ V | = ∆ ( H ) = n ( H ) −
1. Let V ( H u ) ∩ V = { ( u , v ) } and ( u , v ′ ) ∈ V ( H u ) ∩ V . Notice,that v ′ N ( v ) , as f is a γ R ( G ◦ H ) -function. Now, let v ′′ be a universal vertex of H and definea function g ′ as g ′ ( u , v ′′ ) = g ′ ( u , v ) = g ′ ( u , v ′ ) = g ′ ( x , y ) = f ( x , y ) for the remainingvertices. Obviously, g ′ is an RDF on G ◦ H with ω ( g ′ ) < ω ( f ) = γ R ( G ◦ H ) , which is acontradiction. Therefore, ∆ ( H ) = n ( H ) − V = ∅ .Subcase 2.2. ∆ ( H ) = n ( H ) −
2. By Theorem 3.6, γ R ( G ◦ H ) = ζ ( G ) , and since W ∈ ℘ o ( G ) ∩ D p ( G ) , we have that ( W \ W ′ , W ′ ) is a dominating couple, which implies that γ R ( G ◦ H ) = ζ ( G ) ≤ | W \ W ′ | + | W ′ | = | W | + | W ′ | ≤ | V | + | V | = γ R ( G ◦ H ) , which implies that ( W \ W ′ , W ′ ) is a ζ ( G ) -couple.Now, assume δ ( H ) ≥
1. Suppose that there exists x ∈ W \ W ′ , and let ( x , y ) ∈ V . In sucha case, ( V ( H x ) \ { ( x , y ) } ) ⊆ V , which implies that N ( x ) ∩ W = ∅ , but this is a contradictionas deg ( y ) ≤ n ( H ) −
2. Thus, W = W ′ .Subcase 2.3. ∆ ( H ) = n ( H ) −
3. Assume first that δ ( H ) =
0. By Theorem 4.3, for any ζ ′ ( G ) -set S = S ∪ S we have γ pR ( G ◦ H ) ≤ | S | + | S | = ζ ′ ( G ) . Now, from Theorem 3.6 we have that γ R ( G ◦ H ) = γ t ( G ) , and since W ∈ ℘ o ( G ) ∩ D p ( G ) and f ( H x ) = x ∈ W ′ , wehave that 2 γ t ( G ) = γ R ( G ◦ H ) = γ pR ( G ◦ H ) ≤ ζ ′ ( G ) ≤ | W \ W ′ | + | W ′ | ≤ γ R ( G ◦ H ) = γ t ( G ) .Therefore, ζ ′ ( G ) = γ t ( G ) .On the other side, if δ ( H ) ≥
1, then we can proceed as in Subcase 2.2 to deduce that W = W ′ is a packing and also a perfect dominating set of G , which implies that γ p ( G ) = γ ( G ) = ρ ( G ) . Thus, f ( H x ) = x ∈ W , and so by Theorem 3.6 we have that2 γ t ( G ) = γ R ( G ◦ H ) = γ pR ( G ◦ H ) = | W | = γ p ( G ) = ρ ( G ) . Therefore, γ t ( G ) = γ p ( G ) = ρ ( G ) .Subcase 2.4. ∆ ( H ) ≤ n ( H ) −
4. In this case, | V ( H u ) ∩ V | ≥
3. Hence, for any u ′ ∈ N ( u ) and v ∈ V ( H ) , the function g ′ , defined by g ′ ( H u ) = g ′ ( H u ′ ) = g ′ ( u , v ) = g ′ ( u ′ , v ) = g ′ ( x , y ) = f ( x , y ) for the remaining vertices, is an RDF on G ◦ H of weight ω ( g ′ ) < ω ( f ) = γ R ( G ◦ H ) ,which is again a contradiction. Therefore, ∆ ( H ) ≤ n ( H ) − V = ∅ .We proceed to summarize the conclusions derived from the cases above, and to prove thestatements.Proof of (i). Assume ∆ ( H ) = n ( H ) −
1. As we have shown in Case 1 and Subcase 2.1, from γ pR ( G ◦ H ) = γ R ( G ◦ H ) we deduce that P ( G , H ) holds.Conversely, if ∆ ( H ) = n ( H ) − P ( G , H ) holds, then by Theorems 4.12-(i), 3.2 and3.6, γ pR ( G ◦ H ) = γ p ( G ◦ H ) = γ ( G ) = γ R ( G ◦ H ) . Therefore, (i) follows.Proof of (ii). Assume ∆ ( H ) = n ( H ) −
2. If γ pR ( G ◦ H ) = γ R ( G ◦ H ) , then we have to considerCase 1 and Subcase 2.2.From Case 1, γ R ( G ◦ H ) = γ t ( G ) and P ( G , H ) holds. Thus, δ ( H ) = S of G with | S | = γ t ( G ) . Since ( S , ∅ ) is a dominating coupleand 2 | S | = γ t ( G ) = γ R ( G ◦ H ) , by Theorem 3.6 we conclude that ( S , ∅ ) is a ζ ( G ) -couple and,obviously, S ∈ ℘ o ( G ) .On the other hand, in Subcase 2.2 we concluded that ( W \ W ′ , W ′ ) is a ζ ( G ) -couple and W ∈ ℘ o ( G ) . Also, W = W ′ whenever δ ( H ) ≥ ( A , B ) be a ζ ( G ) -couple such that A ∪ B ∈ ℘ o ( G ) . Let v ∈ V ( H ) be a vertexof maximum degree and let { v ′ } = V ( H ) \ N [ v ] .Notice that if v ′ is an isolated vertex, then the function g ( X , X , X ) , defined by X = A × { v ′ } ∪ B × { v } and X = B × { v ′ } , is a PRDF on G ◦ H . Hence, by Theorem 3.6, ζ ( G ) = γ R ( G ◦ H ) ≤ γ pR ( G ◦ H ) ≤ ω ( g ) = | A | + | B | = ζ ( G ) . Therefore, γ pR ( G ◦ H ) = γ R ( G ◦ H ) .Now, if deg ( v ′ ) ≥ A = ∅ , then B is a packing and also a dominating set, whichimplies that the function g ( X , X , X ) , defined by X = B × { v } and X = B × { v ′ } , is a PRDFon G ◦ H . Hence, by Theorem 3.6, ζ ( G ) = γ R ( G ◦ H ) ≤ γ pR ( G ◦ H ) ≤ ω ( g ) = | B | = ζ ( G ) .Therefore, γ pR ( G ◦ H ) = γ R ( G ◦ H ) , as required.Proof of (iii). Assume ∆ ( H ) = n ( H ) −
3. If γ pR ( G ◦ H ) = γ R ( G ◦ H ) , then we have to considerCase 1 and Subcase 2.3.In Case 1 we deduced that δ ( H ) = γ pR ( G ◦ H ) = | W | = γ t ( G ) . Furthermore, since P ( G , H ) holds, W ∈ ℘ o ( G ) ∩ D p ( G ) . Thus, ζ ′ ( G ) ≤ | W | and, by Theorem 4.3, 2 | W | = γ pR ( G ◦ H ) ≤ ζ ′ ( G ) ≤ | W | , which implies that ζ ′ ( G ) = γ t ( G ) .In Subcase 2.3, we deduced that if δ ( H ) =
0, then ζ ′ ( G ) = γ t ( G ) , while if δ ( H ) ≥ γ t ( G ) = γ p ( G ) = ρ ( G ) .Conversely, assume that ζ ′ ( G ) = γ t ( G ) and δ ( H ) =
0. Let S = S ∪ S be a ζ ′ ( G ) -set.By Theorems 4.3 and 3.6 we have that 2 γ t ( G ) = γ R ( G ◦ H ) ≤ γ pR ( G ◦ H ) ≤ | S | + | S | = ζ ′ ( G ) = γ t ( G ) , which implies that γ R ( G ◦ H ) = γ pR ( G ◦ H ) .Now, assume δ ( H ) ≥ γ t ( G ) = γ p ( G ) = ρ ( G ) . Let v ∈ V ( H ) be a vertex of maxi-mum degree, { v , v } = V ( H ) \ N [ v ] and D a γ p ( G ) -set. Notice that the function g ( X , X , X ) ,defined by X = D × { v } and X = D × { v , v } , is a PRDF on G ◦ H . Hence, by Theorem3.6 we have that 2 γ t ( G ) = γ R ( G ◦ H ) ≤ γ pR ( G ◦ H ) ≤ ω ( g ) = | X | + | X | = γ p ( G ) = γ t ( G ) .Thus, γ pR ( G ◦ H ) = γ R ( G ◦ H ) , which completes the proof of (iii).Proof of (iv). Assume ∆ ( H ) ≤ n ( H ) −
4. As shown in Case 1 and Subcase 2.4, from γ pR ( G ◦ H ) = γ R ( G ◦ H ) we deduce that P ( G , H ) holds. Conversely, if P ( G , H ) holds, then by The-orems 4.12-(iii), 3.2 and 3.6 it follows that γ pR ( G ◦ H ) = γ p ( G ◦ H ) = γ t ( G ) = γ R ( G ◦ H ) .Therefore, (iv) follows. References [1] H. Abdollahzadeh Ahangar, M. A. Henning, V. Samodivkin, I. G. Yero, Total Romandomination in graphs, Appl. Anal. Discrete Math. 10 (2016) 501–517.[2] B. H. Arriola, S. R. Canoy, Jr., Doubly connected domination in the corona and lexico-graphic product of graphs, Appl. Math. Sci. (Ruse) 8 (29-32) (2014) 1521–1533.[3] S. Banerjee, J. M. Keil, D. Pradhan, Perfect Roman domination in graphs, Theoret. Com-put. Sci. 796 (2019) 1–21.[4] A. Cabrera Mart´ınez, S. Cabrera Garc´ıa, J. A. Rodr´ıguez-Vel´azquez, Double dominationin lexicographic product graphs, Discrete Appl. Math. 284 (2020) 290–300.175] A. Cabrera Mart´ınez, A. Estrada-Moreno, J. A. Rodr´ıguez-Vel´azquez, From Italian dom-ination in lexicographic product graphs to ww