𝑘-Tuple Total Restrained Domination in Complementary Prisms

Abstract

In a graph 𝐺 with 𝛿(𝐺)≥𝑘≥1, a 𝑘-tuple total restrained dominating set 𝑆 is a subset of 𝑉(𝐺) such that each vertex of  𝑉(𝐺) is adjacent to at least 𝑘 vertices of 𝑆 and also each vertex of 𝑉(𝐺)−𝑆 is adjacent to at least 𝑘 vertices of  𝑉(𝐺)−𝑆. The minimum number of vertices of such sets in 𝐺 is the 𝑘-tuple total restrained domination number of 𝐺. In [𝑘-tuple total restrained domination/domatic in graphs, BIMS], the author initiated the study of the 𝑘-tuple total restrained domination number in graphs. In this paper, we continue it in the complementary prism of a graph.

1. Introduction

Let 𝐺 be a simple graph with the vertex set  𝑉=𝑉(𝐺) and the edge set 𝐸=𝐸(𝐺). The order  |𝑉| and size  |𝐸| of 𝐺 are denoted by 𝑛=𝑛(𝐺) and 𝑚=𝑚(𝐺), respectively. The open neighborhood and the closed neighborhood of a vertex 𝑣∈𝑉 are 𝑁_𝐺(𝑣)={𝑢∈𝑉∣𝑢𝑣∈𝐸} and 𝑁_𝐺[𝑣]=𝑁_𝐺(𝑣)∪{𝑣}, respectively. Also the open neighborhood and the closed neighborhood of a subset 𝑋⊆𝑉(𝐺) are 𝑁_𝐺(𝑋)=∪_𝑣∈𝑋𝑁_𝐺(𝑣) and 𝑁_𝐺[𝑋]=𝑁_𝐺(𝑋)∪𝑋, respectively. The degree of a vertex 𝑣∈𝑉 is deg(𝑣)=|𝑁(𝑣)|. The minimum and maximum degree of a graph 𝐺 are denoted by 𝛿=𝛿(𝐺) and Δ=Δ(𝐺), respectively. If every vertex of 𝐺 has degree 𝑘, then 𝐺 is called 𝑘-regular. We write 𝐾_𝑛, 𝐶_𝑛, and 𝑃_𝑛 for a complete graph, a cycle and a path of order 𝑛, respectively, while 𝐾_{𝑛1,…,𝑛𝑝} denotes a complete 𝑝-partite graph. The complement of a graph 𝐺 is denoted by 𝐺 and is a graph with the vertex set 𝑉(𝐺) and for every two vertices 𝑣 and 𝑤, 𝑣𝑤∈𝐸(𝐺) if and only if 𝑣𝑤∉𝐸(𝐺).

For each integer 𝑘≥1, the 𝑘-join  𝐺∘_𝑘𝐻 of a graph 𝐺 to a graph 𝐻 of order at least 𝑘 is the graph obtained from the disjoint union of 𝐺 and 𝐻 by joining each vertex of 𝐺 to at least 𝑘 vertices of 𝐻 [1]. Also, 𝐺∘_∗𝑘𝐻 denotes the 𝑘-join 𝐺∘_𝑘𝐻 such that each vertex of 𝐺 is joined to exact 𝑘 vertices of 𝐻.

The complementary prism  𝐺𝐺 of 𝐺 is the graph formed from the disjoint union 𝐺∪𝐺 of 𝐺 and 𝐺 by adding the edges of a perfect matching between the corresponding vertices (same label) of 𝐺 and 𝐺 [2]. For example, the graph 𝐶₅𝐶₅ is the Petersen graph. Also, if 𝐺=𝐾_𝑛, the graph 𝐾_𝑛𝐾_𝑛 is the corona 𝐾_𝑛∘𝐾₁, where the corona  𝐺∘𝐾₁ of a graph 𝐺 is the graph obtained from 𝐺 by attaching a pendant edge to each vertex of 𝐺.

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 𝑘-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 𝑘≥1 be an integer and let 𝐺 be a graph with 𝛿(𝐺)≥𝑘. A subset 𝑆⊆𝑉(𝐺) is called a 𝑘-tuple total dominating set, briefly kTDS, in 𝐺, if for each 𝑥∈𝑉(𝐺), |𝑁(𝑥)∩𝑆|≥𝑘. The minimum number of vertices of a 𝑘-tuple total dominating set in a graph 𝐺 is called the 𝑘-tuple total domination number of 𝐺 and denoted by 𝛾_×𝑘,𝑡(𝐺).
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 𝑑(𝐺) and the total domatic number 𝑑_𝑡(𝐺) of a graph were introduced in [7, 8], respectively. Sheikholeslami and Volkmann extended the last definition to the 𝑘-tuple total domatic number 𝑑_×𝑘,𝑡(𝐺) in [9] and Kazemi extended it to the star 𝑘-tuple total domatic number 𝑑^∗_×𝑘,𝑡(𝐺) in [10].

Definition 2. The 𝑘-tuple total domatic partition, briefly kTDP, of 𝐺 is a partition 𝔻 of the vertex set of 𝐺 such that all classes of 𝔻 are 𝑘-tuple total dominating sets in 𝐺. The maximum number of classes of a 𝑘-tuple total domatic partition of 𝐺 is called the 𝑘-tuple total domatic number 𝑑_×𝑘,𝑡(𝐺) of 𝐺 [9]. The star  𝑘-tuple total domatic number  𝑑^∗_×𝑘,𝑡(𝐺) of 𝐺 is the maximum number of classes of a kTDP of 𝐺 such that at least one of the 𝑘-tuple total dominating sets in it has cardinality 𝛾_×𝑘,𝑡(𝐺) [10].
The author in [10] initiated the study of the 𝑘-tuple total restrained domination number and the 𝑘-tuple total restrained domatic number of graphs.

Definition 3 (see [10]). The 𝑘-tuple total restrained domatic partition, briefly kTRDP, of 𝐺 is a partition 𝔻 of the vertex set of 𝐺 such that all classes of 𝔻 are 𝑘-tuple total restrained dominating sets in 𝐺. The maximum number of classes of a 𝑘-tuple total restrained domatic partition of 𝐺 is the 𝑘-tuple total restrained domatic number  𝑑^𝑟_×𝑘,𝑡(𝐺) of 𝐺. Similarly, the star  𝑘-tuple total restrained domatic number  𝑑^𝑟∗_×𝑘,𝑡(𝐺) of 𝐺 is the maximum number of classes of a kTRDP of 𝐺 such that at least one of the 𝑘-tuple total restrained dominating sets in it has cardinality 𝛾^𝑟_×𝑘,𝑡(𝐺).
In this paper, we continue our studies, which is initiated in [10] and find some sharp bounds for the 𝑘-tuple total restrained domination number of the complementary prism of a graph. Also we will find the 𝑘-tuple total restrained domination number of a cycle, a path, and a complete multipartite graph.
Through 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)={𝑣∣𝑣∈𝑉(𝐺)} and 𝐸(𝐺𝐺)=𝐸(𝐺)∪𝐸(𝐺)∪{𝑣𝑣∣𝑣∈𝑉(𝐺)} such that 𝐸(𝐺)={𝑢𝑣∣𝑢𝑣∉𝐸(𝐺)}. The vertices 𝑣 and 𝑣 are called corresponding vertices. Also for a subset 𝑋⊆𝑉(𝐺), we show its corresponding subset in 𝐺 by 𝑋. Also we assume that 𝑉(𝐶_𝑛)=𝑉(𝑃_𝑛)={𝑖∣1≤𝑖≤𝑛}, 𝑉(𝐶_𝑛)=𝑉(𝑃_𝑛)   =  {𝑖∣1≤𝑖≤𝑛}, 𝐸(𝐶_𝑛)   =  𝐸(𝑃_𝑛)∪{1𝑛}  =  {𝑖𝑗∣1≤𝑖≤𝑛−1, and 𝑗=𝑖+1}  ∪  {1𝑛}.
The next known results are useful for our investigations.

Proposition 4 (see Henning and Kazemi [1] 2010). Let 𝑝≥2 be an integer and let 𝐺=𝐾_{𝑛1,𝑛2,…,𝑛𝑝} be a complete 𝑝-partite graph with 𝑛₁≤𝑛₂≤⋯≤𝑛_𝑝.(i)If 𝑘<𝑝, then 𝛾_×𝑘,𝑡(𝐺)=𝑘+1.(ii)If 𝑘=𝑝 and ∑^𝑘−1_𝑖=1𝑛_𝑖≥𝑘, then 𝛾_×𝑘,𝑡(𝐺)=𝑘+2.(iii)If 2≤𝑝<𝑘 and ⌈𝑘/(𝑝−1)⌉≤𝑛₁≤𝑛₂≤⋯≤𝑛_𝑝, then 𝛾_×𝑘,𝑡(𝐺)=⌈𝑘𝑝/(𝑝−1)⌉.

Proposition 5 (see Kazemi [6] 2011). Let 𝑛≥4. Then 𝛾_𝑡𝑃_𝑛𝑃_𝑛=2𝑛−24(),2+1𝑖𝑓𝑛≡3mod4𝑛−24+2𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒. (1)

Proposition 6 (see Kazemi [6] 2011). Let 𝑛≥4. Then 𝛾_𝑡𝐶_𝑛𝐶_𝑛=2𝑛4(),2𝑛+2𝑖𝑓𝑛≡0mod44(),2𝑛+1𝑖𝑓𝑛≡3mod44𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒. (2)

Proposition 7 (see Kazemi [6] 2011). If n  ≥  5, then 𝛾_×2,𝑡(𝐶_𝑛𝐶_𝑛)=𝑛+2.

Proposition 8 (see Kazemi [6] 2011). Let 𝐺=𝐾_{𝑛1,𝑛2,…,𝑛𝑝} be a complete 𝑝-partite graph with 𝑉(𝐺𝐺)=⋃_{1≤𝑖≤𝑝}(𝑋_𝑖∪𝑋_𝑖), when for each 𝑖, 𝑋_𝑖 is isomorph to 𝐾_𝑛𝑖. If 𝑆 is a kTDS of 𝐺𝐺, then for each 1≤𝑖≤𝑝, |𝑆∩𝑋_𝑖|≥𝑘. Furthermore, if |𝑆∩𝑋_𝑖|=𝑘 for some 𝑖, then |𝑆∩𝑋_𝑖|≥𝑘.

Proposition 9 (see Kazemi [6] 2011). Let 𝐺=𝐾_{𝑛1,𝑛2,…,𝑛𝑝} be a complete 𝑝-partite graph with 1≤𝑛₁≤𝑛₂≤⋯≤𝑛_𝑝. Then 𝛾_𝑡𝐺𝐺=2𝑝−𝛼, (3) where 𝛼=|{𝑖∣1≤𝑖≤𝑝, and 𝑛_𝑖=1}|.

Proposition 10 (see Kazemi [6] 2011). Let 𝐺=𝐾_{𝑛1,𝑛2,…,𝑛𝑝} be a complete 𝑝-partite graph with 3≤𝑛₁≤𝑛₂≤⋯≤𝑛_𝑝. Then 𝛾_×2,𝑡(𝐺𝐺)=3𝑝+2.

Proposition 11 (see Kazemi [6] 2011). Let 𝐺=𝐾_{𝑛1,𝑛2,…,𝑛𝑝} be a complete 𝑝-partite graph with 5≤𝑘+2≤𝑛₁≤⋯≤𝑛_𝑝. Then 𝛾_×𝑘,𝑡𝐺𝐺=()()𝑝()𝑝+1𝑘+1𝑖𝑓𝑝≥𝑘+1,𝑝+1𝑘+1+1𝑖𝑓𝑝=𝑘≥4,16𝑖𝑓𝑝=𝑘=3,𝑘+1+min2𝑘−2,𝑘𝑝𝑝−1𝑖𝑓𝑝<𝑘. (4)

Proposition 12 (see Kazemi [10] 2011). Let 𝐺 be a graph of order 𝑛 in which 𝛿(𝐺)≥𝑘. Then(i)every vertex of degree at most 2𝑘−1 of 𝐺 and at least its 𝑘 neighbors belong to every kTRDS;(ii)𝑑^𝑟_×𝑘,𝑡(𝐺)=1 if 𝛿(𝐺)≤2𝑘−1;(iii)Δ(𝐺)≥2𝑘 if 𝛾^𝑟_×𝑘,𝑡(𝐺)<𝑛. Hence, 𝑛≥2𝑘+2;(iv)𝛾_×𝑘,𝑡(𝐺)≤𝛾^𝑟_×𝑘,𝑡(𝐺), and so 𝑑^𝑟_×𝑘,𝑡(𝐺)≤𝑑_×𝑘,𝑡(𝐺).

Proposition 13 (see Kazemi [10] 2011). Let 𝐺 be a graph with minimum degree at least 𝑘. If 𝑑^∗_×𝑘,𝑡(𝐺)≥2, then 𝛾^𝑟_×𝑘,𝑡(𝐺)=𝛾_×𝑘,𝑡(𝐺).

Proposition 14 (see Kazemi [10] 2011). Let 𝑛≥𝑘+3≥4. Then 𝛾^𝑟_×𝑘,𝑡𝐶_𝑛=𝑛𝑖𝑓𝑛≤2𝑘+2,𝑘+2𝑖𝑓2𝑘+3≤𝑛≤3𝑘+2,𝑘+1𝑖𝑓𝑛≥3𝑘+3. (5)

Proposition 15 (see Kazemi [10] 2011). Let 𝑘<𝑛 be positive integers. Then 𝛾^𝑟_×𝑘,𝑡𝐾_𝑛=𝑛𝑖𝑓𝑛≤2𝑘+1,𝑘+1𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒. (6)

Proposition 16 (see Kazemi [10] 2011). Let 𝑛≥4. Then 𝛾^𝑟_𝑡𝐶_𝑛=2𝑛4(),2𝑛−1𝑖𝑓𝑛≡1mod44(),2𝑛+1𝑖𝑓𝑛≡3mod44𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒. (7)

2. Some Bounds

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

Theorem 17. Let 𝑘 and ℓ be integers such that 1≤𝑘−1≤ℓ≤2𝑘−2. If 𝐺 is a ℓ-regular graph of order 𝑛, then 𝛾^𝑟_×𝑘,𝑡𝐺𝐺≥𝑛+𝑘, (8) with equality if and only if 𝑛≥ℓ+2𝑘 and 𝑉(𝐺) contains a 𝑘-subset 𝑇 such that for each vertex 𝑖∈𝑉(𝐺), |𝑁(𝑖)∩𝑇|≥𝑘−1 and also if 𝑖∈𝑉(𝐺)−𝑇, then |𝑁(𝑖)∩(𝑉(𝐺)−𝑇)|≥𝑘.

Proof. Let 𝑉(𝐺𝐺)=𝑉(𝐺)∪𝑉(𝐺) such that 𝑉(𝐺)={𝑖∣1≤𝑖≤𝑛} and 𝑉(𝐺)={𝑖∣1≤𝑖≤𝑛}. Let 𝑛≥2𝑘+ℓ, and let 𝑆 be an arbitrary kTRDS of 𝐺𝐺. Then Proposition 12(i) and this fact that every vertex 𝑖∈𝑉(𝐺) has degree ℓ+1≤2𝑘−1 imply that 𝑉(𝐺)⊆𝑆. Let 𝑖∉𝑆. Then |𝑁(𝑖)∩𝑉(𝐺)∩𝑆|≥𝑘−1. If |𝑁(𝑖)∩𝑉(𝐺)∩𝑆|≥𝑘, then we have nothing to prove. Thus, let 𝑁(𝑖)∩𝑉(𝐺)∩𝑆={𝑗_𝑚∣1≤𝑚≤𝑘−1}. But, this implies that there exists a vertex 𝑡∈𝑆−{𝑗_𝑚∣1≤𝑚≤𝑘−1} such that its corresponding vertex 𝑡 in 𝑉(𝐺) is adjacent to some vertex 𝑗_𝑚, when 1≤𝑚≤𝑘−1. Then |𝑆|≥𝑛+𝑘, and since 𝑆 was arbitrary, we conclude that 𝛾^𝑟_×𝑘,𝑡(𝐺𝐺)≥𝑛+𝑘.
Obviously, it can be seen that 𝛾^𝑟_×𝑘,𝑡(𝐺𝐺)=𝑛+𝑘 if and only if 𝑛≥ℓ+2𝑘 and 𝑉(𝐺) contains a 𝑘-subset 𝑇 such that for each vertex 𝑖∈𝑉(𝐺), |𝑁(𝑖)∩𝑇|≥𝑘−1 and also if 𝑖∈𝑉(𝐺)−𝑇, then |𝑁(𝑖)∩(𝑉(𝐺)−𝑇)|≥𝑘.

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

Corollary 18. Let 𝑘 and ℓ be integers such that 1≤𝑘−1≤ℓ≤2𝑘−2. If 𝐺 is a ℓ-regular graph of order 𝑛≤ℓ+2𝑘−1, then 𝛾^𝑟_×𝑘(𝐺𝐺)=2𝑛.

The next theorem gives lower and upper bounds for 𝛾^𝑟_×𝑘,𝑡(𝐺𝐺), when 𝐺 is an arbitrary graph.

Theorem 19. If 𝐺 is a graph of order 𝑛 with 𝑘≤min{𝛿(𝐺),𝛿(𝐺)}, then 𝛾^𝑟_{×()𝑘−1,𝑡}(𝐺)+𝛾^𝑟_{×()𝑘−1,𝑡}𝐺≤𝛾^𝑟_×𝑘,𝑡𝐺𝐺≤𝛾^𝑟_×𝑘,𝑡(𝐺)+𝛾^𝑟_×𝑘,𝑡𝐺, (9) when 𝑘≥1 in the upper bound and 𝑘≥2 in the lower bound.

Proof. To prove 𝛾^𝑟_{×(𝑘−1),𝑡}(𝐺)+𝛾^𝑟_{×(𝑘−1),𝑡}(𝐺)≤𝛾^𝑟_×𝑘,𝑡(𝐺𝐺), let 𝑘≥2 and let 𝐷 be a kTRDS of 𝐺𝐺. Since every vertex of 𝑉(𝐺) (resp., 𝑉(𝐺)) is adjacent to only one vertex of 𝑉(𝐺) (resp., 𝑉(𝐺)), we conclude that 𝐷∩𝑉(𝐺) is a (𝑘−1) TRDS of 𝐺 and 𝐷∩𝑉(𝐺) is a (𝑘−1) TRDS of 𝐺. Then 𝛾^𝑟_{×()𝑘−1,𝑡}(𝐺)+𝛾^𝑟_{×()𝑘−1,𝑡}𝐺≤|𝐷∩𝑉(𝐺)|+𝐷∩𝑉𝐺=|𝐷|=𝛾^𝑟_×𝑘,𝑡𝐺𝐺. (10) We now prove that 𝛾^𝑟_×𝑘,𝑡(𝐺𝐺)≤𝛾^𝑟_×𝑘,𝑡(𝐺)+𝛾^𝑟_×𝑘,𝑡(𝐺), when 𝑘≥1. Since for every kTRDS 𝑆 of 𝐺 and every kTRDS 𝑆^′ of 𝐺, the set 𝑆∪𝑆^′ is a kTRDS of 𝐺𝐺, we have 𝛾^𝑟_×𝑘,𝑡𝐺𝐺≤𝛾^𝑟_×𝑘,𝑡(𝐺)+𝛾^𝑟_×𝑘,𝑡𝐺. (11)

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 𝛾^𝑟_×𝑘,𝑡(𝐺𝐺) when 𝐺 is a cycle, a path, or a complete multipartite graph. First, let 𝐺 be a cycle.

Proposition 20. Let 𝑛≥4. Then 𝛾^𝑟_×2,𝑡𝐶_𝑛𝐶_𝑛=2𝑛𝑖𝑓𝑛=4,5,𝑛+2𝑖𝑓𝑛≥6. (12)

Proof. Corollary 18 implies that 𝛾^𝑟_×2,𝑡(𝐶_𝑛𝐶_𝑛)=2𝑛 if 𝑛=4,5. Now let 𝑛≥6. Proposition 7 with Proposition 12(iv) implies 𝛾^𝑟_×2,𝑡(𝐶_𝑛𝐶_𝑛)≥𝑛+2. Since also 𝑉(𝐶_𝑛)∪{1,4} is a 2TRDS of 𝐶_𝑛𝐶_𝑛, we get 𝛾^𝑟_×2,𝑡(𝐶_𝑛𝐶_𝑛)=𝑛+2 when 𝑛≥6.

To calculate 𝛾_𝑡(𝐶_𝑛𝐶_𝑛) we need to prove that 𝑑^∗_𝑡(𝐶_𝑛𝐶_𝑛)≥2.

Proposition 21. Let 𝑛≥4. Then 𝑑^∗_𝑡(𝐶_𝑛𝐶_𝑛)≥2.

Proof. We prove the proposition in the following four cases.
Case 1 (𝑛≡0(mod4)). For 𝑛=4, we set 𝑆={1,1,2,2} and 𝑆^′={3,3,4,4}. If 𝑛>4, we set 𝑆={1,1,2,2}∪{5+4𝑖,6+4𝑖∣0≤𝑖≤⌈𝑛/4⌉−2} and 𝑆^′ = {3,3,4,4}  ∪  {7+4𝑖,8+4𝑖∣0 ≤ 𝑖≤⌈𝑛/4⌉−2}.
Case 2  (𝑛≡1(mod4)). For 𝑛=5, we set 𝑆={1,1,4,4} and 𝑆^′={2,2,5,5} and for 𝑛=9, we set 𝑆 = {1,1,4,4,7,7} and 𝑆^′ = {2,2,5,5,8,8}. If 𝑛>9, we set 𝑆 = {1,1,4,4,7,7}  ∪  {10+4𝑖,11+4𝑖∣0 ≤ 𝑖≤⌈𝑛/4⌉−4} and  𝑆^′={3,3,6,6,9,9}  ∪  {12+4𝑖,13+4𝑖∣0≤𝑖 ≤ ⌈𝑛/4⌉−4}.
Case 3 (𝑛≡2(mod4)). For 𝑛=6, we set 𝑆={1,1,4,4} and 𝑆^′={2,2,5,5}. If 𝑛>6, we set 𝑆={1,1,4,4}  ∪  {7+4𝑖,8+4𝑖∣0≤𝑖≤⌈𝑛/4⌉−3} and 𝑆^′={3,3,6,6}  ∪  {9+4𝑖,10+4𝑖∣0≤𝑖≤⌈𝑛/4⌉−3}.
Case 4  (𝑛≡3(mod4)). For 𝑛=7, we set 𝑆={1,1,4,4,6} and 𝑆^′={2,2,5,5,7}. If 𝑛>7, we set 𝑆={1,1,4,4,𝑛−1}  ∪  {7+4𝑖,8+4𝑖∣0≤𝑖≤⌈𝑛/4⌉−3} and 𝑆^′={2,3,3,6,6}  ∪  {9+4𝑖,10+4𝑖∣0≤𝑖≤⌈𝑛/4⌉−3}.
Since, in all cases, 𝑆 and 𝑆^′ are two disjoint 𝛾_𝑡(𝐶_𝑛𝐶_𝑛)-sets, we have 𝑑^∗_𝑡(𝐶_𝑛𝐶_𝑛)≥2.

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

Proposition 22. Let 𝑛≥4. Then 𝛾^𝑟_𝑡𝐶_𝑛𝐶_𝑛=2𝑛4(),2𝑛+2𝑖𝑓𝑛≡0mod44(),2𝑛+1𝑖𝑓𝑛≡3mod44𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒. (13)

Now we continue our work when 𝐺 is a path.

Proposition 23. Let 𝑛≥4. Then 𝛾^𝑟_𝑡𝑃_𝑛𝑃_𝑛=2𝑛4(),2𝑛+2𝑖𝑓𝑛≡0mod44(),2𝑛+1𝑖𝑓𝑛≡0mod44𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒. (14)

Proof . Propositions 5 and 12(iv) imply that 𝛾^𝑟_𝑡𝑃_𝑛𝑃_𝑛≥𝛾_𝑡𝑃_𝑛𝑃_𝑛=2𝑛−24(),2+1𝑖𝑓𝑛≡0mod4𝑛−24+2otherwise. (15)
Let 𝑛≡0(mod4). For 𝑛 =8, we set 𝑆 ={1,8,3,4,5,6} and for 𝑛>8 we set 𝑆=1,𝑛−6,𝑛−5,∪𝑛𝑛,𝑛−3,𝑛−23+4𝑖,4+4𝑖∣0≤𝑖≤4−3. (16)
If 𝑛≡1,2,3(mod4), then we set 𝑆 ={1,𝑛−2,𝑛,𝑛 −2}  ∪{3+4𝑖,4+4𝑖∣0≤𝑖≤⌊𝑛/4⌋−2}, 𝑆 ={1,𝑛}  ∪  {3+4𝑖,4+4𝑖∣0≤𝑖≤⌊𝑛/4⌋−1}, and 𝑆 ={1,𝑛−1,𝑛}∪{3+4𝑖,4+4𝑖∣0≤𝑖≤⌊𝑛/4⌋−1}, respectively. Since, in each case, 𝑆 is a TRDS of 𝑃_𝑛𝑃_𝑛 of cardinality 𝛾_𝑡(𝑃_𝑛𝑃_𝑛), we have completed our proof.

In the next propositions, we calculate 𝛾^𝑟_×𝑘,𝑡(𝐺𝐺) when 𝐺 is a complete multipartite graph.

Proposition 24. If 𝐺=𝐾_{𝑛1,𝑛2,…,𝑛𝑝} is a complete 𝑝-partite graph, then 𝑑_𝑡𝐺𝐺=min𝑛_𝑖∣1≤𝑖≤𝑝. (17)

Proof. Let 𝑉(𝐺𝐺)=⋃_{1≤𝑖≤𝑝}(𝑋_𝑖∪𝑋_𝑖) and let ℓ=min{𝑛_𝑖∣1≤𝑖≤𝑝}. Proposition 8 implies that for every TDS 𝐷 of 𝐺𝐺, |𝐷∩𝑋_𝑖|≥1 when 1≤𝑖≤𝑝. Hence, 𝑑_𝑡(𝐺𝐺)≤ℓ. Now let 𝐷¹, 𝐷²,… and 𝐷^ℓ be ℓ disjoint 2𝑝-sets of 𝑉(𝐺𝐺) such that for every 𝑗=1,2,…,ℓ and every 𝑖=1,2,…,𝑝, |𝐷^𝑗∩𝑋_𝑖|=|𝐷^𝑗∩𝑋_𝑖|=1 and 𝑥∈𝐷^𝑗∩𝑋_𝑖 if and only if 𝑥∈𝐷^𝑗∩𝑋_𝑖. Since 𝐷¹, 𝐷², …, and 𝐷^ℓ are ℓ disjoint 𝛾_𝑡-sets of 𝐺𝐺, by Proposition 9, we get 𝑑_𝑡(𝐺𝐺)≥ℓ, and so 𝑑_𝑡(𝐺𝐺)=ℓ.

Proposition 25. If 𝐺=𝐾_{𝑛1,…,𝑛𝑝} is a complete 𝑝-partite graph with 2≤𝑛₁≤⋯≤𝑛_𝑝, then 𝛾^𝑟_𝑡(𝐺𝐺)=2𝑝.

Proof. Since obviously 𝑑_𝑡(𝐺𝐺) =𝑑^∗_𝑡(𝐺𝐺), we obtain 𝛾^𝑟_𝑡(𝐺𝐺)=2𝑝, by Propositions 9, 13, and 24.

Proposition 26. Let 𝐺=𝐾_𝑛1,𝑛2 be a complete bipartite graph with 4≤𝑛₁≤𝑛₂. Then 𝛾^𝑟_×2,𝑡(𝐺𝐺)=8.

Proof. Let 𝑆 be a set of vertices such that for 𝑖=1,2, |𝑆∩𝑋_𝑖|=|𝑆∩𝑋_𝑖|=2 and 𝑥∈𝑆∩𝑋_𝑖 if and only if 𝑥∈𝑆∩𝑋_𝑖. Since 𝑆 is a 2TRDS of 𝐺𝐺 of cardinality 𝛾_×2,𝑡(𝐺𝐺), we get 𝛾^𝑟_×2,𝑡(𝐺𝐺)=8, by Propositions 10 and 12(iv).

Proposition 27. Let 𝐺=𝐾_{𝑛1,𝑛2,…,𝑛𝑝} be a complete 𝑝-partite graph with 5≤𝑛₁≤⋯≤𝑛_𝑝. Then 𝛾^𝑟_×2,𝑡(𝐺𝐺)=3𝑝+2.

Proof. Let 𝑆 be a set of vertices such that for 𝑖=1,2, |𝑆∩𝑋_𝑖|=|𝑆∩𝑋_𝑖|=2 and 𝑥∈𝑆∩𝑋_𝑖 if and only if 𝑥∈𝑆∩𝑋_𝑖, and for 𝑖=3,…,𝑝,  |𝑆∩𝑋_𝑖|=3. Obviously 𝑆 is a 𝛾_×2,𝑡(𝐺𝐺)-set. It can easily be verified that every vertex in out of 𝑆 is adjacent to at least two vertices in out of 𝑆. Hence, 𝑆 is a 2TRDS of 𝐺𝐺 of cardinality 𝛾_×2,𝑡(𝐺𝐺), and so 𝛾^𝑟_×2,𝑡(𝐺𝐺)=3𝑝+2, by Propositions 10 and 12(iv).

Proposition 28. Let 𝐺 =𝐾_{𝑛1,𝑛2,𝑛3} be a complete 3-partite graph with 6≤𝑛₁≤𝑛₂≤𝑛₃. Then 𝛾^𝑟_×3,𝑡(𝐺𝐺)=16.

Proof. Let 𝑆 be a set of vertices such that |𝑆∩𝑋₃|=4, and for 𝑖=1,2, |𝑆∩𝑋_𝑖|=|𝑆∩𝑋_𝑖|=3 such that 𝑥∈𝑆∩𝑋_𝑖 if and only if 𝑥∈𝑆∩𝑋_𝑖. Since 𝑆 is a 3TRDS of 𝐺𝐺 of cardinality 𝛾_×3,𝑡(𝐺𝐺), we get 𝛾^𝑟_×3,𝑡(𝐺𝐺)=16, by Propositions 11 and 12(iv).

Proposition 29. Let 𝐺=𝐾_{𝑛1,𝑛2,…,𝑛𝑝} be a complete 𝑝-partite graph with 7≤2𝑘+1≤𝑛₁≤⋯≤𝑛_𝑝. Then 𝛾^𝑟_×𝑘,𝑡𝐺𝐺=()()𝑝()𝑝+1𝑘+1𝑖𝑓𝑝≥𝑘+1,𝑝+1𝑘+1+1𝑖𝑓𝑝=𝑘≥4,16𝑖𝑓𝑝=𝑘=3,𝑘+1+min2𝑘−2,𝑘𝑝𝑝−1𝑖𝑓𝑝<𝑘. (18)

Proof. We prove the proposition in the following three cases.
Case 1. (𝑝≥𝑘+1or 𝑝=𝑘≥4). Let 𝑆 be a set of vertices such that 𝑆∩𝑉(𝐺) is a 𝛾_×𝑘,𝑡(𝐺)-set and for 𝑖=1,2,…,𝑝, |𝑆∩𝑋_𝑖|=𝑘+1. Obviously, 𝑆 is a 𝛾_×𝑘,𝑡(𝐺𝐺)-set. It can easily be verified that every vertex in out of 𝑆 is adjacent to at least 𝑘 vertices in out of 𝑆. Hence, 𝑆 is a kTRDS of 𝐺𝐺 of cardinality 𝛾_×𝑘,𝑡(𝐺𝐺), and so 𝛾^𝑟_×𝑘,𝑡(𝐺𝐺) is (𝑝+1)(𝑘+1) if 𝑝≥𝑘+1 and is (𝑝+1)(𝑘+1)+1 if 𝑝=𝑘≥4, by Propositions 11 and 12(iv).
Case 2. (𝑝=𝑘=3). This case is proved in Proposition 28 when 6=2𝑘≤𝑛₁≤𝑛₂≤𝑛₃.
Case 3.  (𝑝<𝑘). If min{2𝑘−2,⌈𝑘𝑝/(𝑝−1)⌉}=2𝑘−2, let 𝑆 be a set of vertices such that for 𝑖=1,2, |𝑆∩𝑋_𝑖|=|𝑆∩𝑋_𝑖|=𝑘 and 𝑥∈𝑆∩𝑋_𝑖 if and only if 𝑥∈𝑆∩𝑋_𝑖, and for 𝑖=3,…,𝑝,|𝑆∩𝑋_𝑖|=𝑘+1. If min{2𝑘−2,⌈𝑘𝑝/(𝑝−1)⌉}=⌈𝑘𝑝/(𝑝−1)⌉, let 𝑆 be a set of vertices such that 𝑆∩𝑉(𝐺) is a 𝛾_×𝑘,𝑡(𝐺)-set and for 𝑖=1,2,…,𝑝,  |𝑆∩𝑋_𝑖|=𝑘+1. In all cases, obviously 𝑆 is a 𝛾_×𝑘,𝑡(𝐺𝐺)-set, and it can easily be verified that every vertex in out of 𝑆 is adjacent to at least 𝑘 vertices in out of 𝑆. Hence, 𝑆 is a kTRDS of 𝐺𝐺 of cardinality 𝛾_×𝑘,𝑡(𝐺𝐺), and so 𝛾^𝑟_×2,𝑡(𝐺𝐺)=𝑝(𝑘+1)+min{2𝑘−2,⌈𝑘𝑝/(𝑝−1)⌉}, 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 𝑘-tuple total restrained domination number of a complete multipartite graph and its complement.

Proposition 30. Let 𝐺=𝐾_{𝑛1,𝑛2,…,𝑛𝑝} be a complete 𝑝-partite graph of order 𝑛 with 𝑘+1≤𝑛₁≤⋯≤𝑛_𝑡≤2𝑘+1<𝑛_𝑡+1≤⋯≤𝑛_𝑝. If 𝑝≥𝑘+1, then 𝛾^𝑟_×𝑘,𝑡𝐺=𝑛₁+⋯+𝑛_𝑡+𝑝−𝑡(𝑘+1). (19)

Proof. Since 𝐺 is the union of disjoint complete graphs 𝐾_𝑛1, 𝐾_𝑛2,…, 𝐾_𝑛𝑝, Proposition 15 implies that 𝛾^𝑟_×𝑘,𝑡𝐺=𝛾^𝑟_×𝑘,𝑡𝐾_𝑛1+𝛾^𝑟_×𝑘,𝑡𝐾_𝑛2+⋯𝛾^𝑟_×𝑘,𝑡𝐾_𝑛𝑝=𝑛₁+⋯+𝑛_𝑡+𝑝−𝑡(𝑘+1). (20)

Proposition 31. Let 𝐺=𝐾_{𝑛1,𝑛2,…,𝑛𝑝} be a complete 𝑝-partite graph with 2≤𝑛₁≤⋯≤𝑛_𝑝. If 𝑝≥𝑘+1, then 𝛾^𝑟_×𝑘,𝑡(𝐺)=𝑘+1.

Proof. Let 𝑆 be a set of vertices of cardinality 𝑘+1 such that for every index 𝑖=1,2,…,𝑘+1, |𝑆∩𝑋_𝑖|=1. Since 𝑆 is a kTRDS of 𝐺 of cardinality 𝛾_×𝑘,𝑡(𝐺), Propositions 12(iv) and 4(i) imply that 𝛾^𝑟_×𝑘,𝑡(𝐺)=𝑘+1.

Proposition 32. Let 𝐺=𝐾_{𝑛1,𝑛2,…,𝑛𝑘} be a complete 𝑘-partite graph of order 𝑛 with 𝑛₁≤𝑛₂≤⋯≤𝑛_𝑘. If 𝑛−𝑛_𝑘−1≥𝑛−𝑛_𝑘≥2𝑘 and 𝑛−𝑛₁≥⋯≥𝑛−𝑛_𝑘−2≥2𝑘+1, then 𝛾^𝑟_×𝑘,𝑡(𝐺)=𝑘+2.

Proof. Let 𝑆 be a set of vertices of cardinality 𝑘+2 such that |𝑆∩𝑋_𝑖|=2 for 𝑖=𝑘−1,𝑘, and |𝑆∩𝑋_𝑖|=1 for 𝑖=1,2,…,𝑘−2. Since 𝑆 is a kTRDS of 𝐺 of cardinality 𝛾_×𝑘,𝑡(𝐺), Propositions 4(ii) and 12(iv) imply that 𝛾^𝑟_×𝑘,𝑡(𝐺)=𝑘+2.

Proposition 33. Let 𝐺=𝐾_{𝑛1,𝑛2,…,𝑛𝑝} be a complete 𝑝-partite graph of order with ⌈𝑘/(𝑝−1)⌉≤𝑛₁≤⋯≤𝑛_𝑝, and let 𝑝<𝑘. If 𝑝−1∣𝑘 and, for each index 𝑖=1,2,…,𝑝, we have 𝑛−𝑛_𝑖≥2𝑘, then 𝛾^𝑟_×𝑘,𝑡(𝐺)=𝑘𝑝/(𝑝−1).

Proof. Let 𝑆 be a set of vertices of cardinality 𝑘𝑝/(𝑝−1) such that for each 𝑖=1,2,…,𝑝, |𝑆∩𝑋_𝑖|=𝑘/(𝑝−1). Since 𝑆 is a kTRDS of 𝐺 of cardinality 𝛾_×𝑘,𝑡(𝐺), Propositions 4(iii) and 12(iv) imply that 𝛾^𝑟_×𝑘,𝑡(𝐺)=𝑘𝑝/(𝑝−1).

Proposition 34. Let 𝐺=𝐾_{𝑛1,𝑛2,…,𝑛𝑝} be a complete 𝑝-partite graph of order 𝑛 with ⌈𝑘/(𝑝−1)⌉≤𝑛₁≤⋯≤𝑛_𝑝, and let 𝑝<𝑘. Let also 𝑘 = (𝑝−1)⋅⌊𝑘/(𝑝−1)⌋+𝑟 for some integer 1≤𝑟≤𝑝−2. If 𝑛−𝑛_𝑝−𝑟 ≥ 𝑛−𝑛_{𝑝−𝑟+1}≥⋯≥𝑛−𝑛_𝑝 ≥ ⌈𝑘𝑝/(𝑝−1)⌉−⌈𝑘/(𝑝−1)⌉+𝑘  and  𝑛−𝑛₁≥⋯≥𝑛−𝑛_{𝑝−𝑟−1} ≥ ⌈𝑘𝑝/(𝑝−1)⌉−⌈𝑘/(𝑝−1)⌉+𝑘+1, then 𝛾^𝑟_×𝑘,𝑡(𝐺)=⌈𝑘𝑝/(𝑝−1)⌉.

Proof. Let 𝑆 be a set of vertices of cardinality 1≤𝑖≤𝑝−𝑟−1, |𝑆∩𝑋_𝑖|=⌊𝑘/(𝑝−1)⌋ and for 𝑝−𝑟≤𝑖≤𝑝,|𝑆∩𝑋_𝑖|=⌈𝑘/(𝑝−1)⌉=⌊𝑘/(𝑝−1)⌋+1. Since 𝑆 is a kTRDS of 𝐺 of cardinality 𝛾_×𝑘,𝑡(𝐺), Propositions 4(iii) and 12(iv) imply that 𝛾^𝑟_×𝑘,𝑡(𝐺)=⌈𝑘𝑝/(𝑝−1)⌉.

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 𝑘≥1, let 𝐺=𝐾_{𝑛1,𝑛2,…,𝑛𝑝} be a complete 𝑝-partite graph of order 𝑛 with 2𝑘+2≤𝑛₁≤𝑛₂≤⋯≤𝑛_𝑝. If either 𝑝≥𝑘+1 or 𝑝=𝑘≥4 and 2𝑘≤𝑛−𝑛_𝑘≤𝑛−𝑛_𝑘−1, 2𝑘+1≤𝑛−𝑛_𝑘−2≤⋯≤𝑛−𝑛₁, then 𝛾^𝑟_×𝑘,𝑡𝐺𝐺=𝛾^𝑟_×𝑘,𝑡(𝐺)+𝛾^𝑟_×𝑘,𝑡𝐺. (21)

Proposition 36. Let 𝐺=𝐾_{𝑛1,𝑛2,…,𝑛𝑝} be a complete 𝑝-partite graph, and let 𝑘≥1 be an integer. If 𝑘≤𝑛₁≤⋯≤𝑛_𝑡≤2𝑘−1<𝑛_𝑡+1≤⋯≤𝑛_𝑝 and 𝑛₁+𝑛₂+⋯+𝑛_𝑡=𝑝+1+𝑡𝑘 for some integer 𝑡≥1, then 𝛾^𝑟_×𝑘,𝑡𝐺𝐺=𝛾^𝑟_{×()𝑘−1,𝑡}(𝐺)+𝛾^𝑟_{×()𝑘−1,𝑡}𝐺. (22)

We note that there are some complete multipartite graphs that satisfy Proposition 36. For example, if 𝑘≥4, then the complete (𝑘+1)-partite graph 𝐺=𝐾_{𝑛1,𝑛2,…,𝑛𝑘+1} with the conditions 𝑛₁=𝑘+3, 𝑛₂=2𝑘−1, and 2𝑘≤𝑛₃≤⋯≤𝑛_𝑘+1 satisfies it.

Also Propositions 14, 16, and 20 by this fact that 𝛾^𝑟_×2,𝑡(𝐶_𝑛)=𝑛, which is obtained by Proposition 12(i), imply that 𝛾^𝑟_×2,𝑡𝐶_𝑛𝐶_𝑛=𝛾^𝑟_×2,𝑡𝐶_𝑛+𝛾^𝑟_×𝑘,𝑡𝐶_𝑛 (23) if and only if 𝑛=5. Also they imply that 𝛾^𝑟_𝑡𝐶_𝑛+𝛾^𝑟_𝑡𝐶_𝑛<𝛾^𝑟_×2,𝑡𝐶_𝑛𝐶_𝑛<𝛾^𝑟_×2,𝑡𝐶_𝑛+𝛾^𝑟_×𝑘,𝑡𝐶_𝑛, (24) when 𝑛≥4 in the lower bound and 𝑛≥6 in the upper bound.

5. Open Problem

Finally, we finish our paper with the following problems.

Problem 37. Characterize graphs 𝐺 that satisfy 𝛾^𝑟_×𝑘,𝑡(𝐺𝐺)=𝛾^𝑟_×𝑘,𝑡(𝐺)+𝛾^𝑟_×𝑘,𝑡(𝐺).

Problem 38. Characterize graphs 𝐺 that satisfy 𝛾^𝑟_×𝑘,𝑡(𝐺𝐺)=𝛾^𝑟_{×(𝑘−1),𝑡}(𝐺)+𝛾^𝑟_{×(𝑘−1),𝑡}(𝐺).