Abstract

Let 𝑘 be a positive integer, and let 𝐺 be a graph with minimum degree at least 𝑘. In their study (2010), Henning and Kazemi defined the 𝑘-tuple total domination number 𝛾×𝑘,𝑡(𝐺) of 𝐺 as the minimum cardinality of a 𝑘-tuple total dominating set of 𝐺, which is a vertex set such that every vertex of 𝐺 is adjacent to at least 𝑘 vertices in it. If 𝐺 is the complement of 𝐺, the complementary prism 𝐺𝐺 of 𝐺 is the graph formed from the disjoint union of 𝐺 and 𝐺 by adding the edges of a perfect matching between the corresponding vertices of 𝐺 and 𝐺. In this paper, we extend some of the results of Haynes et al. (2009) for the 𝑘-tuple total domination number and also obtain some other new results. Also we find the 𝑘-tuple total domination number of the complementary prism of a cycle, a path, or a complete multipartite graph.

1. Introduction

In this paper, 𝐺=(𝑉,𝐸) is a simple graph with the vertex set 𝑉 and the edge set 𝐸. The order |𝑉| of 𝐺 is denoted by 𝑛=𝑛(𝐺). The open neighborhood and the closed neighborhood of a vertex 𝑣𝑉 are 𝑁𝐺(𝑣)={𝑢𝑉(𝐺)𝑢𝑣𝐸(𝐺)} and 𝑁𝐺[𝑣]=𝑁𝐺(𝑣){𝑣}, respectively. Also the degree of 𝑣 is deg𝐺(𝑣)=|𝑁𝐺(𝑣)|. Similarly, the open neighborhood and the closed neighborhood of a set 𝑆𝑉 are 𝑁𝐺(𝑆)=𝑣𝑆𝑁(𝑣) and 𝑁𝐺[𝑆]=𝑁𝐺(𝑆)𝑆, respectively. The complement of 𝐺 is the graph 𝐺 with the vertex set 𝑉(𝐺)=𝑉(𝐺) and the edge set 𝐸(𝐺)={𝑢𝑣𝑢𝑣𝐸(𝐺)}. The minimum and maximum degree of 𝐺 are denoted by 𝛿=𝛿(𝐺) and Δ=Δ(𝐺), respectively. We also write 𝐾𝑛, 𝐶𝑛, and 𝑃𝑛 for the complete graph, cycle, and path of order 𝑛, respectively, while 𝐺[𝑆] and 𝐾𝑛1,𝑛2,,𝑛𝑝 denote the subgraph induced on 𝐺 by a vertex set 𝑆, and the complete 𝑝-partite graph, respectively.

Haynes et al. in [1] have defined complementary product of two graphs that generalizes the Cartesian product of two graphs. Let 𝐺 and 𝐻 be two graphs. For each 𝑅𝑉(𝐺) and 𝑆𝑉(𝐻), the complementary product 𝐺(𝑅)𝐻(𝑆) is a graph with the vertex set {(𝑢𝑖,𝑣𝑗)𝑢𝑖𝑉(𝐺),𝑣𝑖𝑉(𝐻)} and (𝑢𝑖,𝑣𝑗)(𝑢,𝑣𝑘) is an edge in 𝐸(𝐺(𝑅)𝐻(𝑆))(1)if 𝑖=, 𝑢𝑖𝑅, and 𝑣𝑗𝑣𝑘𝐸(𝐻) or if 𝑖=, 𝑢𝑖𝑅, and 𝑣𝑗𝑣𝑘𝐸(𝐻), or(2)if 𝑗=𝑘, 𝑣𝑗𝑆, and 𝑢𝑖𝑢𝐸(𝐺) or if 𝑗=𝑘, 𝑣𝑗𝑆, and 𝑢𝑖𝑢𝐸(𝐺).

In other words, for each 𝑢𝑖𝑉(𝐺), we replace 𝑢𝑖 by a copy of 𝐻 if 𝑢𝑖 is in 𝑅 and by a copy of its complement 𝐻 if 𝑢𝑖 is not in 𝑅, and for each 𝑣𝑗𝑉(𝐻), we replace each 𝑣𝑗 by a copy of 𝐺 if 𝑣𝑗𝑆 and by a copy of 𝐺 if 𝑣𝑗𝑆. If 𝑅=𝑉(𝐺) (resp., 𝑆=𝑉(𝐻)), we write simply 𝐺𝐻(𝑆) (resp., 𝐺(𝑅)𝐻). Thus, 𝐺𝐻(𝑆) is the graph obtained by replacing each vertex 𝑣 of 𝐻 by a copy of 𝐺 if 𝑣𝑆 and by a copy of 𝐺 if 𝑣𝑆 and replacing each vertex 𝑢 of 𝐺 by a copy of 𝐻. We recall that the Cartesian product 𝐺𝐻 of two graphs 𝐺 and 𝐻 is the complementary product 𝐺(𝑉(𝐺))𝐻(𝑉(𝐻)). The special complementary product 𝐺𝐾2(𝑆), where |𝑆|=1, is called the complementary prism of 𝐺 and denoted by 𝐺𝐺. For example, the graph 𝐶5𝐶5 is the Petersen graph. Also, if 𝐺=𝐾𝑛, the graph 𝐾𝑛𝐾𝑛 is the corona 𝐾𝑛𝐾1, where the corona 𝐺𝐾1 of a graph 𝐺 is the graph obtained from 𝐺 by attaching a pendant edge to each vertex of 𝐺. We notice that 𝛿(𝐺𝐺)=min{𝛿(𝐺),𝛿(𝐺)}+1.

In [2], Henning and Kazemi introduced the 𝑘-tuple total domination number of graphs. Let 𝑘 be a positive integer. A subset 𝑆 of 𝑉 is a 𝑘-tuple total dominating set of 𝐺, abbreviated kTDS, if for every vertex 𝑣𝑉, |𝑁(𝑣)𝑆|𝑘, that is, 𝑆 is a kTDS of 𝐺 if every vertex of 𝑉 has at least 𝑘 neighbors in 𝑆. The 𝑘-tuple total domination number 𝛾×𝑘,𝑡(𝐺) of 𝐺 is the minimum cardinality of a kTDS of 𝐺. We remark that a 1-tuple total domination is the well-studied total domination number. Thus, 𝛾𝑡(𝐺)=𝛾×1,𝑡(𝐺). For a graph to have a 𝑘-tuple total dominating set, its minimum degree is at least 𝑘. Since every (𝑘+1)-tuple total dominating set is also a 𝑘-tuple total dominating set, we note that 𝛾×𝑘,𝑡(𝐺)𝛾×(𝑘+1),𝑡(𝐺) for all graphs with minimum degree at least 𝑘+1. A kTDS of cardinality 𝛾×𝑘,𝑡(𝐺) is called a 𝛾×𝑘,𝑡(𝐺)-set. When 𝑘=2, a 2-tuple total dominating set is called a double total dominating set, abbreviated DTDS, and the 2-tuple total domination number is called the double total domination number. The redundancy involved in 𝑘-tuple total domination makes it useful in many applications. The paper in [3] gives more information about the 𝑘-tuple total domination number of a graph.

In [4], Haynes et al. discussed the domination and total domination number of complementary prisms. In this paper, we extend some of their results for the 𝑘-tuple total domination number and obtain some other results. More exactly, we find some useful lower and upper bounds for the 𝑘-tuple total domination number of the complementary prism 𝐺𝐺 in terms on the order of 𝐺, 𝛾×𝑘,𝑡(𝐺), 𝛾×𝑘,𝑡(𝐺), 𝛾×(𝑘1),𝑡(𝐺), and 𝛾×(𝑘1),𝑡(𝐺), in which some of the bounds are sharp. Also we find this number for 𝐺𝐺, when 𝐺 is a cycle, a path, or a complete multipartite graph.

Through of this paper, 𝑘 is a positive integer, and for simplicity, we assume that 𝑉(𝐺𝐺) is the disjoint union 𝑉(𝐺)𝑉(𝐺) with 𝑉(𝐺)={𝑣𝑣𝑉(𝐺)} and 𝐸(𝐺𝐺)=𝐸(𝐺)𝐸(𝐺){𝑣𝑣𝑣𝑉(𝐺)} such that 𝐸(𝐺)={𝑢𝑣𝑢𝑣𝐸(𝐺)}. The vertices 𝑣 and 𝑣 are called the corresponding vertices. Also for a subset 𝑋𝑉(𝐺), we show its corresponding subset in 𝐺 with 𝑋. The next known results are useful for our investigations.

Proposition A (Haynes et al. [2]). If 𝐺 is a path or a cycle of order 𝑛5 such that 𝑛2(mod4) or is the corona graph 𝐾𝑛𝐾1, where 𝑛3, then 𝛾𝑡(𝐺𝐺)=𝛾𝑡(𝐺).

Proposition B (Henning and Kazemi [4]). Let 𝑝2 be an integer, and let 𝐺=𝐾𝑛1,𝑛2,,𝑛𝑝 be a complete 𝑝-partite graph, where 𝑛1𝑛2𝑛𝑝.(i)If 𝑘<𝑝, then 𝛾×𝑘,𝑡(𝐺)=𝑘+1,(ii)if 𝑘=𝑝 and 𝑘1𝑖=1𝑛𝑖𝑘, then 𝛾×𝑘,𝑡(𝐺)=𝑘+2,(iii)if 2𝑝<𝑘 and 𝑘/(𝑝1)𝑛1𝑛2𝑛𝑝, then 𝛾×𝑘,𝑡(𝐺)=𝑘𝑝/(𝑝1).

Proposition C (Henning and Kazemi [5]). Let 𝐺 be a graph of order 𝑛 with 𝛿(𝐺)𝑘. Then 𝛾×𝑘,𝑡(𝐺)max𝑘+1,𝑘𝑛.Δ(𝐺)(1.1)

Proposition D (Henning and Kazemi [5]). Let 𝐺 be a graph of order 𝑛 with 𝛿(𝐺)𝑘, and let 𝑆 be a kTDS of 𝐺. Then for every vertex 𝑣 of degree 𝑘 in 𝐺, 𝑁𝐺(𝑣)𝑆.

2. Some Bounds

The next two theorems state some lower and upper bounds for 𝛾×𝑘,𝑡(𝐺𝐺).

Theorem 2.1. If 𝐺 is a graph of order 𝑛 with 2𝑘min{𝛿(𝐺),𝛿(𝐺)}, then 𝛾×(𝑘1),𝑡(𝐺)+𝛾×(𝑘1),𝑡𝐺𝛾×𝑘,𝑡𝐺𝐺𝛾min×(𝑘1),𝑡(𝐺),𝛾×(𝑘1),𝑡𝐺+𝑛.(2.1)

Proof. Since for every 𝛾×(𝑘1),𝑡(𝐺)-set 𝐷 the set 𝐷𝑉(𝐺) is a kTDS of 𝐺𝐺, we get 𝛾×𝑘,𝑡(𝐺𝐺)𝛾×(𝑘1),𝑡(𝐺)+𝑛. Similarly, we have 𝛾×𝑘,𝑡(𝐺𝐺)𝛾×(𝑘1),𝑡(𝐺)+𝑛. Therefore 𝛾×𝑘,𝑡𝐺𝐺𝛾min×(𝑘1),𝑡(𝐺),𝛾×(𝑘1),𝑡𝐺+𝑛.(2.2)
For proving 𝛾×(𝑘1),𝑡(𝐺)+𝛾×(𝑘1),𝑡(𝐺)𝛾×𝑘,𝑡(𝐺𝐺), let 𝐷 be a kTDS of 𝐺𝐺. Then 𝐷𝑉(𝐺) is a (𝑘1)TDS of 𝐺 and 𝐷𝑉(𝐺) is a (𝑘1)TDS of 𝐺. Since every vertex of 𝑉(𝐺) (resp., 𝑉(𝐺)) is adjacent to only one vertex of 𝑉(𝐺) (resp., 𝑉(𝐺)). Therefore 𝛾×(𝑘1),𝑡(𝐺)+𝛾×(𝑘1),𝑡𝐺||||+|||𝐷𝑉(𝐺)𝐷𝑉𝐺|||=||𝐷||=𝛾×𝑘,𝑡𝐺𝐺.(2.3)

The given bounds in Theorem 2.1 are sharp. Let 𝐺 be a (𝑘1)-regular graph of odd order 𝑛=2𝑘1. Then 𝐺 and 𝐺𝐺 are (𝑘1)- and 𝑘-regular, respectively, and Proposition D implies 𝛾×𝑘,𝑡(𝐺𝐺)=2𝑛 and 𝛾×(𝑘1),𝑡(𝐺)=𝛾×(𝑘1),𝑡(𝐺)=𝑛. Therefore𝛾×(𝑘1),𝑡(𝐺)+𝛾×(𝑘1),𝑡𝐺=𝛾×𝑘,𝑡𝐺𝐺𝛾=min×(𝑘1),𝑡(𝐺),𝛾×(𝑘1),𝑡𝐺+𝑛.(2.4) The Harary graphs 𝐻2𝑚,4𝑚+1 [6] are a family of this kind of graphs. We recall that the Harary graph 𝐻2𝑚,𝑛 is a 2𝑚-regular graph with the vertex set {𝑖1𝑖𝑛} and every vertex 𝑖 is adjacent to the 2𝑚 vertices in the set𝜎𝑖𝑗𝜎𝑖𝑗𝑖+𝑗(mod𝑛)or𝜎𝑖𝑗𝑖𝑗(mod𝑛),for.1𝑗𝑚(2.5)

Theorem 2.2. If 𝐺 is a graph of order 𝑛 with 1𝑘min{𝛿(𝐺),𝛿(𝐺)}, then 𝛾max×𝑘,𝑡(𝐺),𝛾×𝑘,𝑡𝐺𝛾×𝑘,𝑡𝐺𝐺𝛾×𝑘,𝑡(𝐺)+𝛾×𝑘,𝑡𝐺,(2.6) and the lower bound is sharp for 𝑘=1.

Proof. Trivially max{𝛾×𝑘,𝑡(𝐺),𝛾×𝑘,𝑡(𝐺)}𝛾×𝑘,𝑡(𝐺𝐺). Let 𝑆 be a kTDS of 𝐺, and let 𝑆 be a kTDS of 𝐺. Then 𝑆𝑆 is a kTDS of 𝐺𝐺, and so 𝛾×𝑘,𝑡𝐺𝐺𝛾×𝑘,𝑡(𝐺)+𝛾×𝑘,𝑡𝐺.(2.7) Proposition A implies that, if 𝑘=1, then the lower bound is sharp for all paths and cycles of order 𝑛5, where 𝑛2(mod4), and for the corona graph 𝐾𝑛𝐾1, where 𝑛3.

In special case 𝑘=1, we get the following result in [1].

Corollary 2.3 (see [1]). If 𝐺 and 𝐺 have no isolated vertices, then 𝛾max𝑡(𝐺),𝛾𝑡𝐺𝛾𝑡𝐺𝐺𝛾𝑡(𝐺)+𝛾𝑡𝐺.(2.8)

3. The Complementary Prism of Some Graphs

In this section, we calculate the 𝑘-tuple total domination number of the complementary prism 𝐺𝐺, when 𝐺 is a complete multipartite graph, a cycle, or a path. First let 𝐺=𝐾𝑛1,𝑛2,,𝑛𝑝 be a complete 𝑝-partite graph with the vertex partition 𝑉(𝐺)=𝑋1𝑋2𝑋𝑝 such that for each 1𝑖𝑝, |𝑋𝑖|=𝑛𝑖 and 𝑛1𝑛2𝑛𝑝. Then 𝑉(𝐺𝐺)=1𝑖𝑝(𝑋𝑖𝑋𝑖), where 𝑋𝑖 denotes the corresponding set of 𝑋𝑖. Trivially for 𝐺𝐺 to have 𝑘-tuple total domination number we should have 𝑘𝑛1𝑛2𝑛𝑝. In the next five propositions, we calculate this number for the complementary prism of the complete 𝑝-partite graph 𝐺. First we state the following key lemma which has an easy proof that is left to the reader.

Lemma 3.1. Let 𝐺=𝐾𝑛1,𝑛2,,𝑛𝑝 be a complete 𝑝-partite graph with 𝑉(𝐺𝐺)=1𝑖𝑝(𝑋𝑖𝑋𝑖). If 𝑆 is a kTDS of 𝐺𝐺, then for each 1𝑖𝑝, |𝑆𝑋𝑖|𝑘. Furthermore, if |𝑆𝑋𝑖|=𝑘 for some 𝑖, then |𝑆𝑋𝑖|𝑘.

Proposition 3.2. Let 𝐺=𝐾𝑛1,𝑛2,,𝑛𝑝 be a complete 𝑝-partite graph with 1𝑛1𝑛2𝑛𝑝. Then 𝛾𝑡𝐺𝐺=2𝑝𝛼,(3.1) where 𝛼=|{𝑖1𝑖𝑝, and 𝑛𝑖=1}|.

Proof. Let 𝑆 be an arbitrary kTDS of 𝐺𝐺, and let 𝑛1=𝑛2==𝑛𝛼=1<𝑛𝛼+1𝑛𝑝. Proposition D implies that for every 1𝑖𝑝, |𝑆𝑋𝑖|2 or |𝑆𝑋𝑖|=1 and |𝑆𝑋𝑖|1. Also if |𝑋𝑖|=1 and |𝑆𝑋𝑖|=0, it implies |𝑆𝑋𝑖|=1. Therefore |𝑆|𝛼+2(𝑝𝛼)=2𝑝𝛼, and hence 𝛾𝑡(𝐺𝐺)2𝑝𝛼. Now we set 𝐴 as a 𝑝-set such that |𝐴𝑋𝑖|=1, for each 1𝑖𝑝. Since 𝐴{𝑥𝑖𝑥𝑖𝐴and𝛼+1𝑖𝑝} is a TDS of 𝐺 of cardinality 2𝑝𝛼, we get 𝛾𝑡(𝐺𝐺)=2𝑝𝛼.

Corollary 3.3 (see [1]). If 𝑛2, then 𝛾𝑡(𝐾𝑛𝐾𝑛)=𝑛.

Proposition 3.4. If 𝐺=𝐾𝑛1,𝑛2,,𝑛𝑝 is a complete 𝑝-partite graph with 2𝑘=𝑛1==𝑛𝛼<𝑛𝛼+1𝑛𝑝, then 𝛾×𝑘,𝑡𝐺𝐺=𝑝(𝑘+1)+2𝑘2if𝛼=1,𝑝(𝑘+1)+𝛼(𝑘1)otherwise.(3.2)

Proof. We discuss 𝛼.
Case 1 (𝛼2). It follows by 𝛼2 and Lemma 3.1 that, for every 𝑘-tuple total dominating set 𝑆 of 𝐺𝐺, |𝑆𝑋𝑖||𝑆𝑋𝑖|=𝑘 for 1𝑖𝛼 and |𝑆𝑋𝑖|𝑘+1 for 𝛼+1𝑖𝑝. Then 𝛾×𝑘,𝑡𝐺𝐺𝑝(𝑘+1)+𝛼(𝑘1).(3.3) Now we set 𝐷=(1𝑖𝛼(𝑋𝑖𝑋𝑖))(𝛼+1𝑖𝑝𝐷𝑖) such that 𝐷𝑖 is a (𝑘+1)-subset of 𝑋𝑖, for 𝛼+1𝑖𝑝. Since 𝐷 is a kTDS of 𝐺𝐺 of cardinality 𝑝(𝑘+1)+𝛼(𝑘1), we have 𝛾×𝑘,𝑡(𝐺𝐺)=𝑝(𝑘+1)+𝛼(𝑘1).Case 2 (𝛼=1). It follows by 𝛼=1 and Lemma 3.1 that, for every kTDS 𝑆 of 𝐺𝐺, 𝑋1𝑋1 is a subset of 𝑆 and also every vertex of 𝑋1𝑋2𝑋𝑝 is adjacent to at least 𝑘 vertices of 𝑆(𝑋1𝑋1). Thus either |𝑆𝑋𝑖|=𝑘+1 for each 2𝑖𝑝 and 2𝑖𝑝|𝑆𝑋𝑖|𝑘1 or ||𝑆𝑋2||||==𝑆𝑋𝛽||||=𝑘,𝑆𝑋𝛽+1||||==𝑆𝑋𝑝||=𝑘+1,(3.4) for some 2𝛽𝑝. Therefore ||𝑆||min{2𝑘+(𝑘1)+(𝑝1)(𝑘+1),2𝑘+2(𝛽1)𝑘+(𝑝𝛽)(𝑘+1)}=𝑝(𝑘+1)+2(𝑘1).(3.5) Now we set 𝐷=(𝑋1𝑋1)(2𝑖𝑝𝐷𝑖)𝐷0 such that 𝐷𝑖 is a (𝑘+1)-subset of 𝑋𝑖 for 2𝑖𝑝 and 𝐷0 is a (𝑘1)-subset of 𝑉(𝐺) such that |𝐷0𝑋2|==|𝐷0𝑋𝑘|=1. Since 𝐷 is a kTDS of 𝐺𝐺 of cardinality 𝑝(𝑘+1)+2𝑘2, we get 𝛾×𝑘,𝑡(𝐺𝐺)=𝑝(𝑘+1)+2𝑘2.

Now let 𝐺=𝐾𝑛1,𝑛2,...,𝑛𝑝 be a complete 𝑝-partite graph with 3𝑘+1=𝑛1==𝑛𝛼<𝑛𝛼+1𝑛𝑝, and let 𝑆 be a minimal kTDS of 𝐺𝐺. Then |𝑆𝑋𝑖|𝑘, by Lemma 3.1. We notice that if |𝑆𝑋𝑖|𝑘+2, for some 𝑖, then we may improve 𝑆 and obtain another kTDS 𝑆 of cardinality |𝑆| such that |𝑆𝑋𝑖|=𝑘+1 (since every vertex in 𝑋𝑖 (respectively 𝑋𝑖) is adjacent to only one vertex in 𝑋𝑖 (respectively 𝑋𝑖)). Therefore, we may assume that for every minimal kTDS 𝑆 of 𝐺𝐺, we have 𝑘|𝑆𝑋𝑖|𝑘+1.

Now let 𝑆 be a minimal kTDS of 𝐺𝐺, and let 𝐵={𝑖|1𝑖𝑝,|𝑆𝑋𝑖|=𝑘} be a set of cardinality 𝛽. We consider the following two cases.

Case 1 (𝛽0). In this case, if 𝑖𝐵, we have |𝑆𝑋𝑖|=|𝑆𝑋𝑖|=𝑘 such that 𝑥𝑆𝑋𝑖 if and only if 𝑥𝑆𝑋𝑖, and |𝑆𝑋𝑖|=𝑘+1 otherwise. If 𝛽2, then ||𝑆||=𝑝(𝑘+1)+𝛽(𝑘1),(3.6) and if 𝛽=1 and 𝐵={𝑖}, then we have also |𝑆(𝑉(𝐺)𝑋𝑖)|=𝑘. Hence ||𝑆||=𝑝(𝑘+1)+2𝑘1.(3.7) Comparing (3.6), (3.7) shows that for 𝛽0 if 𝑆 is a set of vertices such that 𝑆𝑋𝑖={𝑥𝑖𝑗1𝑗𝑘} and 𝑆𝑋𝑖={𝑥𝑖𝑗𝑥𝑖𝑗𝑆𝑋𝑖} for 𝑖=1,2 and |𝑆𝑋𝑖|=𝑘+1 for 3𝑖𝑝, then 𝑆 is a minimum kTDS of 𝐺𝐺 and ||𝑆||=𝑝(𝑘+1)+2𝑘2.(3.8)

Case 2 (𝛽=0). In this case, for each 1𝑖𝑝 we have |𝑆𝑋𝑖|=𝑘+1. We continue our discussion in the next subcases.
Subcase 1 (𝛼𝑘+1 or 𝛼=𝑘𝑝). Then obviously |𝑆𝑉(𝐺)|𝑘. If for 1𝑖𝑘 we consider |𝑆𝑋𝑖|=1, then 𝑆 is a minimum kTDS of 𝐺𝐺 and ||𝑆||=𝑝(𝑘+1)+𝑘.(3.9)Subcase 2 (𝛼<𝑘𝑝). Then obviously |𝑆𝑉(𝐺)|𝑘+1. If we set 𝑆 such that |𝑆𝑋1|=2, and |𝑆𝑋𝑖|=1 when 2𝑖𝑘, then 𝑆 is a minimum kTDS of 𝐺𝐺 and ||𝑆||=𝑝(𝑘+1)+𝑘.(3.10)Subcase 3 (𝛼=𝑝𝑘1). Then obviously |𝑆𝑉(𝐺)|𝛾×(𝑘1),𝑡(𝐺). If 𝑆𝑉(𝐺) is a 𝛾×(𝑘1),𝑡(𝐺)-set, then 𝑆 is a minimum kTDS of 𝐺𝐺, and Proposition B implies ||𝑆||=(𝑝+1)(𝑘+1)if𝛼=𝑝=𝑘1,𝑝(𝑘+1)+(𝑘1)𝑝𝑝1if𝛼=𝑝<𝑘1.(3.11)Subcase 4 (𝛼<𝑝<𝑘). Then obviously |𝑆𝑉(𝐺)|𝛾×𝑘,𝑡(𝐺). If 𝑆𝑉(𝐺) is a 𝛾×𝑘,𝑡(𝐺)-set, then 𝑆 is a minimum kTDS of 𝐺𝐺, and Proposition B implies ||𝑆||=𝑝(𝑘+1)+𝑘𝑝.𝑝1(3.12)

Now let 𝐺=𝐾𝑛1,𝑛2,...,𝑛𝑝 be a complete 𝑝-partite graph with 4𝑘+2𝑛1𝑛2𝑛𝑝, and let 𝑆 is a minimal kTDS of 𝐺𝐺. In this case, we may similarly assume that 𝑘|𝑆𝑋𝑖|𝑘+1. Also it can be easily seen that if |𝑆𝑋𝑖|=𝑘 for some 𝑖, then equality (3.8) holds. Thus let {𝑖1𝑖𝑝,|𝑆𝑋𝑖|=𝑘}=. Then obviously |𝑆𝑉(𝐺)|𝛾×𝑘,𝑡(𝐺). If we choose a set 𝑆 such that 𝑆𝑉(𝐺) is a 𝛾×𝑘,𝑡(𝐺)-set and |𝑆𝑋𝑖|=𝑘+1 for 1𝑖𝑝, then 𝑆 is a minimum kTDS of 𝐺𝐺, and Proposition B implies ||𝑆||=(𝑝+1)(𝑘+1)if𝑝𝑘+1,(𝑝+1)(𝑘+1)+1if𝑝=𝑘,𝑝(𝑘+1)+𝑘𝑝𝑝1if𝑝<𝑘.(3.13) Comparing (3.9), (3.10), (3.11), (3.12), and (3.13) with (3.8) shows that we have proved the following propositions.

Proposition 3.5. Let 𝐺=𝐾𝑛1,𝑛2,...,𝑛𝑝 be a complete 𝑝-partite graph with 3𝑛1𝑛2𝑛𝑝. Then 𝛾×2,𝑡(𝐺𝐺)=3𝑝+2.

Proposition 3.6. Let 𝐺=𝐾𝑛1,𝑛2,...,𝑛𝑝 be a complete 𝑝-partite graph with 4𝑘+1=𝑛1==𝑛𝛼<𝑛𝛼+1𝑛𝑝. Then 𝛾×𝑘,𝑡𝐺𝐺=𝑝(𝑘+1)+𝑘if𝛼=𝑘𝑝or𝛼𝑘+1(𝑝+1)(𝑘+1)if𝛼<𝑘𝑝or𝛼=𝑝=𝑘1,𝑝(𝑘+1)+min2𝑘2,(𝑘1)𝑝𝑝1if𝛼=𝑝<𝑘1,𝑝(𝑘+1)+min2𝑘2,𝑘𝑝𝑝1if𝛼<𝑝<𝑘.(3.14)

Proposition 3.7. Let 𝐺=𝐾𝑛1,𝑛2,...,𝑛𝑝 be a complete 𝑝-partite graph with 5𝑘+2𝑛1𝑛𝑝. Then 𝛾×𝑘,𝑡𝐺𝐺=(𝑝+1)(𝑘+1)if𝑝𝑘+1(𝑝+1)(𝑘+1)+1if𝑝=𝑘4,16if𝑝𝑝=𝑘=3,(𝑘+1)+min2𝑘2,𝑘𝑝𝑝1if𝑝<𝑘.(3.15)

We now determine the 𝑘-tuple total domination number of the complementary prism 𝐶𝑛𝐶𝑛, where 1𝑘3=𝛿(𝐶𝑛𝐶𝑛). Here we assume that 𝑉(𝐶𝑛𝐶𝑛)=𝑉(𝐶𝑛)𝑉(𝐶𝑛), 𝑉(𝐶𝑛)={𝑖1𝑖𝑛}, and 𝐸(𝐶𝑛)={(𝑖,𝑖+1)1𝑖𝑛}. Proposition D implies that 𝛾×3,𝑡(𝐶𝑛𝐶𝑛)=2𝑛. In many references, for example, in [1], it can be seen that, for 𝑛3,𝛾𝑡𝐶𝑛=2𝑛4if2𝑛𝑛1(mod4),41if𝑛1(mod4),(3.16) and trivially we can prove𝛾𝑡𝐶𝑛=4if3𝑛=4,if2𝑛=5,if𝑛6.(3.17) Hence Theorem 2.1 implies that𝛾𝑡𝐶𝑛+2𝛾×2,𝑡𝐶𝑛𝐶𝑛𝑛+2,(3.18) where 𝑛6, and also Theorem 2.2 implies that𝑛𝛾×2,𝑡𝐶𝑛𝐶𝑛𝑛+𝛾×2,𝑡𝐶𝑛,(3.19) where 𝑛5. In chain (3.19) we need to calculate 𝛾×2,𝑡(𝐶𝑛), which is done by the next proposition.

Proposition 3.8. If C𝑛 is a cycle of order n5, then 𝛾×2,𝑡𝐶𝑛=5if4𝑛=5,if36𝑛8,if𝑛9.(3.20)

Proof. Proposition C implies that 𝛾×2,𝑡(𝐶𝑛)3. If 𝑛9, then, for each 1𝑖𝑛, the set {𝑖,𝑖+3,𝑖+6} is a DTDS of 𝐶𝑛 and so 𝛾×2,𝑡(𝐶𝑛)=3. If 6𝑛8, then it can be easily verified that 𝛾×2,𝑡(𝐶𝑛)4. Now since {1,3,4,6} and {1,2,4,6} are double total dominating sets of 𝐶𝑛, where 𝑛=6 and 𝑛=7,8, respectively, we get 𝛾×2,𝑡(𝐶𝑛)=4. Finally if 𝑛=5, then 𝐶5 is 2-regular and Proposition D implies 𝛾×2,𝑡(𝐶5)=5.

Proposition 3.9. If 𝑛5, then 𝛾×2,𝑡(𝐶𝑛𝐶𝑛)=𝑛+2.

Proof. Let 𝑛5. equalities (3.18), (3.19) and Propositions C and 3.8 imply max𝑛,4𝑛𝑛2𝛾×2,𝑡𝐶𝑛𝐶𝑛𝑛+2.(3.21) If 𝑛=5, then max{𝑛,4𝑛/(𝑛2)}=4𝑛/(𝑛2)=7=𝑛+2, and so 𝛾×2,𝑡(𝐶𝑛𝐶𝑛)=𝑛+2. Thus we assume 𝑛6. Then max{𝑛,4𝑛/(𝑛2)}=𝑛 and hence 𝑛𝛾×2,𝑡(𝐶𝑛𝐶𝑛)𝑛+2. Now let 𝑆 be a 𝛾×2,𝑡(𝐶𝑛𝐶𝑛)-set. If 𝑉(𝐶𝑛)𝑆, then 𝑆=𝑉(𝐶𝑛){𝑥,𝑦}, for some two adjacent vertices 𝑥,𝑦𝑉(𝐶𝑛), and so 𝛾×2,𝑡(𝐶𝑛𝐶𝑛)=𝑛+2. Thus we assume 𝑉(𝐶𝑛)̸𝑆. Without loss of generality, let 3𝑆. Since |𝑆{2,4,3}|2, we continue our proof in the following two cases.Case 1 ({2,4}𝑆). Then 1,5,2,4𝑆. We note that, for every 5𝑖𝑛1, 𝑆{𝑖,𝑖+1}. This implies |𝑆|(𝑛14)+6=𝑛+1, and since 3 must be dominated by 𝑆𝑉(𝐶𝑛), we have |𝑆𝑉(𝐶𝑛)|4. If 𝑛𝑆, then 1𝑆 and so |𝑆|𝑛+1+|{1}|=𝑛+2. Let 𝑛𝑆. If 𝑛𝑆, again |𝑆|𝑛+1+|{𝑛}|=𝑛+2. But 𝑛𝑆 implies 𝑛1𝑆. Let ={𝑖𝑆5𝑖𝑛1and𝑖𝑆}. The condition |𝑆𝑉(𝐶𝑛)|4 implies ||2. Therefore for at least one vertex 5𝑥𝑛1, {𝑥,𝑥+1}𝑆 and hence |𝑆|𝑛+|{𝑥,𝑥+1}|=𝑛+2.Case 2 ({4,3}𝑆 (similarly {2,3}𝑆)). Case 1 implies 2𝑆. Then 1,2,4,1,4,5𝑆. Again we see that, for every 5𝑖𝑛2, 𝑆{𝑖,𝑖+1} and so |𝑆|(𝑛24)+8=𝑛+2.Therefore, in the previous all cases, we proved that 𝛾×2,𝑡(𝐶𝑛𝐶𝑛)𝑛+2 and chain (3.21) implies 𝛾×2,𝑡(𝐶𝑛𝐶𝑛)=𝑛+2.

Corollary 3.10. If 𝑛5, then 𝛾×2,𝑡𝐶𝑛𝐶𝑛=𝛾×2,𝑡𝐶𝑛+𝛾×2,𝑡𝐶𝑛1if𝛾𝑛9,×2,𝑡𝐶𝑛+𝛾×2,𝑡𝐶𝑛2if𝛾6𝑛8,×2,𝑡𝐶𝑛+𝛾×2,𝑡𝐶𝑛3if𝑛=5.(3.22)

Now we determine the exact amount of 𝛾𝑡(𝐶𝑛𝐶𝑛) for 𝑛3. Obviously 𝛾𝑡(𝐶3𝐶3)=|𝑉(𝐶3)|=3. In the next proposition we calculate it when 𝑛4.

Proposition 3.11. Let 𝑛4. Then 𝛾𝑡𝐶𝑛𝐶𝑛=2𝑛4+2if2𝑛𝑛0(mod4),4+1if2𝑛𝑛3(mod4),4otherwise.(3.23)

Proof. Theorem 2.2 with equalities (3.16) and (3.17) implies 4𝛾𝑡𝐶4𝐶46,4𝛾𝑡𝐶5𝐶58,(3.24) and if 𝑛6 and 𝑛1(mod4), then 2𝑛4𝛾𝑡𝐶𝑛𝐶𝑛𝑛24+2,(3.25) and if 𝑛6 and 𝑛1(mod4), then 2𝑛41𝛾𝑡𝐶𝑛𝐶𝑛𝑛24+1.(3.26) If 𝑛=4 and 𝑛=5, then the sets {1,2,1,2} and {1,1,4,4} are total dominating sets of 𝐶𝑛𝐶𝑛, respectively. Hence chain (3.24) implies 𝛾𝑡(𝐶𝑛𝐶𝑛)=4 for 𝑛=4,5. Now we assume 𝑛6. For 𝑛2(mod4), since the sets {1,1,4,4} and {1,1,4,4}{7+4𝑖,8+4𝑖0𝑖𝑛/43} are two total dominating sets of 𝐶𝑛𝐶𝑛 of cardinality 2𝑛/4, where 𝑛=6 and 𝑛>6, respectively, we have 𝛾𝑡(𝐶𝑛𝐶𝑛)=2𝑛/4, by chain (3.25). Now let 𝑛2(mod4). We assume that 𝑆 is a TDS of 𝐶𝑛𝐶𝑛. Obviously 𝑆𝑉(𝐶𝑛). If |𝑆𝑉(𝐶𝑛)|=1 and 𝑆𝑉(𝐶𝑛)={1}, then 1,2,𝑛𝑆, and hence |𝑆𝑋|2|𝑋|/4=2(𝑛5)/4, where 𝑋=𝑉(𝐶𝑛){1,2,3,𝑛1,𝑛}. This implies ||𝑆||=||||2𝑛𝑆𝑋+44+2if2𝑛𝑛0(mod4),4+1if2𝑛𝑛3(mod4),4if𝑛1(mod4).(3.27) Now let |𝑆𝑉(𝐶𝑛)|=𝛼2. If 𝑛0,1(mod4), then ||𝐶𝑆𝑉𝑛||2𝑛𝛼4if2𝑛𝛼(mod4),𝑛𝛼4+1otherwise,(3.28) and if 𝑛3(mod4), then ||𝐶𝑆𝑉𝑛||2𝑛𝛼41if2𝑛𝛼+1(mod4),𝑛𝛼4otherwise.(3.29) It can be calculated that ||𝑆||=||𝐶𝑆𝑉𝑛||2𝑛+𝛼4+2if2𝑛𝑛0(mod4),4+1if2𝑛𝑛3(mod4),4if𝑛1(mod4).(3.30) Then by chains (3.25) and (3.26) we have 𝛾𝑡𝐶𝑛𝐶𝑛𝑛=24+2if𝑛0(mod4),(3.31)2𝑛4𝛾𝑡𝐶𝑛𝐶𝑛𝑛24+1if𝑛1(mod4),(3.32)2𝑛4+1𝛾𝑡𝐶𝑛𝐶𝑛𝑛24+2if𝑛3(mod4).(3.33) If 𝑛1(mod4), then the sets {1,1,4,4,7,7} and {1,1,4,4,7,7}{10+4𝑖,11+4𝑖0𝑖𝑛/44} are total dominating sets of 𝐶𝑛𝐶𝑛 of cardinality 2𝑛/4 when 𝑛=9 and 𝑛>9, respectively. Hence 𝛾𝑡(𝐶𝑛𝐶𝑛)=2𝑛/4, by chain (3.32). If also 𝑛3(mod4), the sets {1,1,4,4,𝑛1} and {1,1,4,4,𝑛1}{7+4𝑖,8+4𝑖0𝑖𝑛/43} are total dominating sets of 𝐶𝑛𝐶𝑛 of cardinality 2𝑛/4+1 when 𝑛=7 and 𝑛>7, respectively. Hence 𝛾𝑡(𝐶𝑛𝐶𝑛)=2𝑛/4+1, by chain (3.33).

Finally we determine the 𝑘-tuple total domination number of the complementary prism 𝑃𝑛𝑃𝑛, where 1𝑘<2=𝛿(𝑃𝑛𝑃𝑛). We recall that 𝑉(𝑃𝑛𝑃𝑛)=𝑉(𝑃𝑛)𝑉(𝑃𝑛), 𝑉(𝑃𝑛)={𝑖1𝑖𝑛}, and 𝐸(𝑃𝑛)={𝑖𝑗1𝑖𝑛1,𝑗=𝑖+1}. In many references, for example, in [1], it can be seen that, for 𝑛2,𝛾𝑡𝑃𝑛=2𝑛4if2𝑛𝑛1(mod4),41if𝑛1(mod4),(3.34) and trivially 𝛾𝑡(𝑃𝑛)=|{1,𝑛}|=2, where 𝑛4. Therefore, by Theorems 2.1 and 2.2, for 𝑛4, we have the following chain:𝛾𝑡𝑃𝑛𝛾𝑡𝑃𝑛𝑃𝑛𝛾𝑡𝑃𝑛+2𝛾×2,𝑡𝑃𝑛𝑃𝑛𝑛+2.(3.35) It can be easily proved that 𝛾𝑡(𝑃𝑛𝑃𝑛)=𝑛, where 𝑛=2,3. Next proposition calculates 𝛾𝑡(𝑃𝑛𝑃𝑛) when 𝑛4.

Proposition 3.12. Let 𝑛4. Then 𝛾𝑡𝑃𝑛𝑃𝑛=2𝑛24+1if2𝑛3(mod4),𝑛24+2otherwise.(3.36)

Proof. Let 𝐷 be a 𝛾𝑡-set of the induced path 𝑃𝑛[𝑉(𝑃𝑛){1,𝑛}] of 𝑃𝑛. Since 𝐷{1,𝑛} is a TDS of 𝑃𝑛𝑃𝑛, we have 𝛾𝑡𝑃𝑛𝑃𝑛|||𝐷1,𝑛|||=2𝑛24+1if2𝑛3(mod4),𝑛24+2otherwise.(3.37) Let 𝑛2(mod4). Then chains (3.34), (3.35), (3.37) imply 𝛾𝑡(𝑃𝑛𝑃𝑛)=2(𝑛2)/4+2. Since 2𝑛/4=2(𝑛2)/4+2. Now let 𝑛2(mod4), and let 𝑆 be a TDS of 𝑃𝑛𝑃𝑛. Obviously 𝑆𝑉(𝑃𝑛). In all cases, (i) |𝑆𝑉(𝑃𝑛)|=1 and 𝑆{1,𝑛}, (ii) |𝑆𝑉(𝑃𝑛)|=1 and 𝑆{1,𝑛}=, and (iii) |𝑆𝑉(𝑃𝑛)|2, then similar to the proof of Proposition 3.11, it can be verified that ||𝑆||2𝑛24+1if2𝑛3(mod4),𝑛24+2otherwise.(3.38) Hence chain (3.37) completes the proof of our proposition.

Propositions 3.11 and 3.12 imply the next result in [1].

Corollary 3.13 (see [1]). If 𝐺{𝑃𝑛,𝐶𝑛} with order 𝑛5, then 𝛾𝑡𝐺𝐺=𝛾𝑡(𝐺)if𝛾𝑛2(mod4),𝑡(𝐺)+2if𝛾𝑛0(mod4),𝑡(𝐺)+1otherwise.(3.39)

4. Problems

If we look carefully at the propositions of Section 3, we obtain the following result.

Proposition 4.1. (i) Let 𝐺 be a cycle or a path of order 𝑛4. Then max{𝛾𝑡(𝐺),𝛾𝑡(𝐺)}=𝛾𝑡(𝐺𝐺) if and only if 𝑛2(mod4).
(ii) Let 𝐺 be a cycle of order 𝑛5 or a path of order 𝑛4. Then 𝛾𝑡(𝐺𝐺)=𝛾𝑡(𝐺)+𝛾𝑡(𝐺) if and only if 𝑛0(mod4).
(iii) Let 𝐶𝑛 be a cycle of order 𝑛5. Then 𝛾max×2,𝑡𝐶𝑛,𝛾×2,𝑡𝐶𝑛<𝛾×2,𝑡𝐶𝑛𝐶𝑛<𝛾×2,𝑡𝐶𝑛+𝛾×2,𝑡𝐶𝑛.(4.1)
(iv) Let 𝐶𝑛 be a cycle of order 𝑛5. Then 𝛾𝑡𝐶𝑛+𝛾𝑡𝐶𝑛<𝛾×2,𝑡𝐶𝑛𝐶𝑛𝛾=𝑛+min𝑡𝐶𝑛,𝛾𝑡𝐶𝑛.(4.2)

Therefore it is natural that we state the following problem.

Problem 1. Characterize graphs 𝐺 with(1)𝛾×𝑘,𝑡(𝐺𝐺)=𝛾×𝑘,𝑡(𝐺)+𝛾×𝑘,𝑡(𝐺), (2)𝛾×𝑘,𝑡(𝐺)=max{𝛾×𝑘,𝑡(𝐺),𝛾×𝑘,𝑡(𝐺)}.