#### Abstract

In a graph with , a -tuple total restrained dominating set is a subset of such that each vertex of   is adjacent to at least vertices of and also each vertex of is adjacent to at least vertices of  . The minimum number of vertices of such sets in is the -tuple total restrained domination number of . In [-tuple total restrained domination/domatic in graphs, BIMS], the author initiated the study of the -tuple total restrained domination number in graphs. In this paper, we continue it in the complementary prism of a graph.

#### 1. Introduction

Let be a simple graph with the vertex set   and the edge set . The order   and size   of are denoted by and , respectively. The open neighborhood and the closed neighborhood of a vertex are and , respectively. Also the open neighborhood and the closed neighborhood of a subset are and , respectively. The degree of a vertex is . The minimum and maximum degree of a graph are denoted by and , respectively. If every vertex of has degree , then is called -regular. We write , , and for a complete graph, a cycle and a path of order , respectively, while denotes a complete -partite graph. The complement of a graph is denoted by and is a graph with the vertex set and for every two vertices and , if and only if .

For each integer , the -join   of a graph to a graph of order at least is the graph obtained from the disjoint union of and by joining each vertex of to at least vertices of [1]. Also, denotes the -join such that each vertex of is joined to exact vertices of .

The complementary prism   of is the graph formed from the disjoint union of and by adding the edges of a perfect matching between the corresponding vertices (same label) of and [2]. For example, the graph is the Petersen graph. Also, if , the graph is the corona , where the corona   of a graph is the graph obtained from by attaching a pendant edge to each vertex of .

The research of domination in graphs is an evergreen area of graph theory. Its basic concept is the dominating set. The literature on this subject has been surveyed and detailed in the two books by Haynes et al. [3, 4]. And many variants of the dominating set were introduced and the corresponding numerical invariants were defined for them. For example, the -tuple total dominating set is defined in [1] by Henning and Kazemi, which is an extension of the total dominating set (for more information see [5, 6]).

Definition 1 (see [1]). Let be an integer and let be a graph with . A subset is called a -tuple total dominating set, briefly kTDS, in , if for each , . The minimum number of vertices of a -tuple total dominating set in a graph is called the -tuple total domination number of and denoted by .
A numerical invariant of a graph which is in a certain sense dual to it is the domatic number of a graph. The domatic number and the total domatic number of a graph were introduced in [7, 8], respectively. Sheikholeslami and Volkmann extended the last definition to the -tuple total domatic number in [9] and Kazemi extended it to the star -tuple total domatic number in [10].

Definition 2. The -tuple total domatic partition, briefly kTDP, of is a partition of the vertex set of such that all classes of are -tuple total dominating sets in . The maximum number of classes of a -tuple total domatic partition of is called the -tuple total domatic number of [9]. The star  -tuple total domatic number   of is the maximum number of classes of a kTDP of such that at least one of the -tuple total dominating sets in it has cardinality [10].
The author in [10] initiated the study of the -tuple total restrained domination number and the -tuple total restrained domatic number of graphs.

Definition 3 (see [10]). The -tuple total restrained domatic partition, briefly kTRDP, of is a partition of the vertex set of such that all classes of are -tuple total restrained dominating sets in . The maximum number of classes of a -tuple total restrained domatic partition of is the -tuple total restrained domatic number   of . Similarly, the star  -tuple total restrained domatic number   of is the maximum number of classes of a kTRDP of such that at least one of the -tuple total restrained dominating sets in it has cardinality .
In this paper, we continue our studies, which is initiated in [10] and find some sharp bounds for the -tuple total restrained domination number of the complementary prism of a graph. Also we will find the -tuple total restrained domination number of a cycle, a path, and a complete multipartite graph.
Through this paper, is a positive integer, and for simplicity, we assume that is the disjoint union with and such that . The vertices and are called corresponding vertices. Also for a subset , we show its corresponding subset in by . Also we assume that ,     ,          , and     .
The next known results are useful for our investigations.

Proposition 4 (see Henning and Kazemi [1] 2010). Let be an integer and let be a complete -partite graph with .(i)If , then .(ii)If and , then .(iii)If and , then .

Proposition 5 (see Kazemi [6] 2011). Let . Then

Proposition 6 (see Kazemi [6] 2011). Let . Then

Proposition 7 (see Kazemi [6] 2011). If n  ≥  5, then .

Proposition 8 (see Kazemi [6] 2011). Let be a complete -partite graph with , when for each , is isomorph to . If is a kTDS of , then for each , . Furthermore, if for some , then .

Proposition 9 (see Kazemi [6] 2011). Let be a complete -partite graph with . Then where , and .

Proposition 10 (see Kazemi [6] 2011). Let be a complete -partite graph with . Then .

Proposition 11 (see Kazemi [6] 2011). Let be a complete -partite graph with . Then

Proposition 12 (see Kazemi [10] 2011). Let be a graph of order in which . Then(i)every vertex of degree at most of and at least its neighbors belong to every kTRDS;(ii) if ;(iii) if . Hence, ;(iv), and so .

Proposition 13 (see Kazemi [10] 2011). Let be a graph with minimum degree at least . If , then .

Proposition 14 (see Kazemi [10] 2011). Let . Then

Proposition 15 (see Kazemi [10] 2011). Let be positive integers. Then

Proposition 16 (see Kazemi [10] 2011). Let . Then

#### 2. Some Bounds

We first give a sharp lower bound for the -tuple total restrained domination number of a regular graph.

Theorem 17. Let and be integers such that . If is a -regular graph of order , then with equality if and only if and contains a -subset such that for each vertex , and also if , then .

Proof. Let such that and . Let , and let be an arbitrary kTRDS of . Then Proposition 12(i) and this fact that every vertex has degree imply that . Let . Then . If , then we have nothing to prove. Thus, let . But, this implies that there exists a vertex such that its corresponding vertex in is adjacent to some vertex , when . Then , and since was arbitrary, we conclude that .
Obviously, it can be seen that if and only if and contains a -subset such that for each vertex , and also if , then .

Theorem 17 and Proposition 12(i) imply the next corollary.

Corollary 18. Let and be integers such that . If is a -regular graph of order , then .

The next theorem gives lower and upper bounds for , when is an arbitrary graph.

Theorem 19. If is a graph of order with , then when in the upper bound and in the lower bound.

Proof. To prove , let and let be a kTRDS of . Since every vertex of (resp., ) is adjacent to only one vertex of (resp., ), we conclude that is a TRDS of and is a TRDS of . Then We now prove that , when . Since for every kTRDS of and every kTRDS of , the set is a kTRDS of , we have

In Section 4, we will show that the given bounds in Theorem 19 are sharp.

#### 3. The Complementary Prism of Some Graphs

In this section, we will determine when is a cycle, a path, or a complete multipartite graph. First, let be a cycle.

Proposition 20. Let . Then

Proof. Corollary 18 implies that if . Now let . Proposition 7 with Proposition 12(iv) implies . Since also is a 2TRDS of , we get when .

To calculate we need to prove that .

Proposition 21. Let . Then .

Proof. We prove the proposition in the following four cases.
Case 1 . For , we set and . If , we set and =     .
Case 2  . For , we set and and for , we set = and = . If , we set =      and      .
Case 3. For , we set and . If , we set      and     .
Case 4  . For , we set and . If , we set      and     .
Since, in all cases, and are two disjoint -sets, we have .

By Propositions 6, 13, and 21 we obtain the next result.

Proposition 22. Let . Then

Now we continue our work when is a path.

Proposition 23. Let . Then

Proof . Propositions 5 and 12(iv) imply that
Let . For  =, we set  = and for we set
If , then we set  = −  ,  =    , and  =, respectively. Since, in each case, is a TRDS of of cardinality , we have completed our proof.

In the next propositions, we calculate when is a complete multipartite graph.

Proposition 24. If is a complete -partite graph, then

Proof. Let and let . Proposition 8 implies that for every TDS of , when . Hence, . Now let , ,… and be disjoint -sets of such that for every and every , and if and only if . Since , , …, and are disjoint -sets of , by Proposition 9, we get , and so .

Proposition 25. If is a complete -partite graph with , then .

Proof. Since obviously  =, we obtain , by Propositions 9, 13, and 24.

Proposition 26. Let be a complete bipartite graph with . Then .

Proof. Let be a set of vertices such that for , and if and only if . Since is a 2TRDS of of cardinality , we get , by Propositions 10 and 12(iv).

Proposition 27. Let be a complete -partite graph with . Then .

Proof. Let be a set of vertices such that for , and if and only if , and for ,  . Obviously is a -set. It can easily be verified that every vertex in out of is adjacent to at least two vertices in out of . Hence, is a 2TRDS of of cardinality , and so , by Propositions 10 and 12(iv).

Proposition 28. Let  = be a complete 3-partite graph with . Then .

Proof. Let be a set of vertices such that , and for , such that if and only if . Since is a 3TRDS of of cardinality , we get , by Propositions 11 and 12(iv).

Proposition 29. Let be a complete -partite graph with . Then

Proof. We prove the proposition in the following three cases.
Case 1. or . Let be a set of vertices such that is a -set and for , . Obviously, is a -set. It can easily be verified that every vertex in out of is adjacent to at least vertices in out of . Hence, is a kTRDS of of cardinality , and so is if and is if , by Propositions 11 and 12(iv).
Case 2. . This case is proved in Proposition 28 when .
Case 3.  . If , let be a set of vertices such that for , and if and only if , and for ,. If , let be a set of vertices such that is a -set and for ,  . In all cases, obviously is a -set, and it can easily be verified that every vertex in out of is adjacent to at least vertices in out of . Hence, is a kTRDS of of cardinality , and so , by Propositions 11 and 12(iv).

#### 4. The Given Bounds in Theorem 19 Are Sharp

In this section, we show that the given bounds in Theorem 19 are sharp. For this aim, we first calculate the -tuple total restrained domination number of a complete multipartite graph and its complement.

Proposition 30. Let be a complete -partite graph of order with . If , then

Proof. Since is the union of disjoint complete graphs , ,…, , Proposition 15 implies that

Proposition 31. Let be a complete -partite graph with . If , then .

Proof. Let be a set of vertices of cardinality such that for every index , . Since is a kTRDS of of cardinality , Propositions 12(iv) and 4(i) imply that .

Proposition 32. Let be a complete -partite graph of order with . If and , then .

Proof. Let be a set of vertices of cardinality such that for , and for . Since is a kTRDS of of cardinality , Propositions 4(ii) and 12(iv) imply that .

Proposition 33. Let be a complete -partite graph of order with , and let . If and, for each index , we have , then .

Proof. Let be a set of vertices of cardinality such that for each , . Since is a kTRDS of of cardinality , Propositions 4(iii) and 12(iv) imply that .

Proposition 34. Let be a complete -partite graph of order with , and let . Let also = for some integer . If   and  , then .

Proof. Let be a set of vertices of cardinality , and for ,. Since is a kTRDS of of cardinality , Propositions 4(iii) and 12(iv) imply that .

Comparing Propositions 29, 30, 31, 32, 33, and 34 shows that the following two results, which state the given bounds in Theorem 19 are sharp, are proved.

Proposition 35. For every integer , let be a complete -partite graph of order with . If either or and , , then

Proposition 36. Let be a complete -partite graph, and let be an integer. If and for some integer , then

We note that there are some complete multipartite graphs that satisfy Proposition 36. For example, if , then the complete ()-partite graph with the conditions , , and satisfies it.

Also Propositions 14, 16, and 20 by this fact that , which is obtained by Proposition 12(i), imply that if and only if . Also they imply that when in the lower bound and in the upper bound.

#### 5. Open Problem

Finally, we finish our paper with the following problems.

Problem 37. Characterize graphs that satisfy .

Problem 38. Characterize graphs that satisfy .