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2-Domination Number in Special Classes of Graphs

1*

2

, S. Muthukumar

3

1*Department of Mathematics, Easwari Engineering College, Chennai, India. E-mail: [email protected]

2Department of Science and Humanities, SaveethaSchool of Engineering, SaveethaInstitute of Medical andTechnical Sciences, Saveetha University, Chennai, India. E-mail: [email protected]

3Department of Mathematics, Easwari Engineering College, Chennai, India. E-mail: [email protected]

ABSTRACT

A set 𝑆 ⊆ 𝑉is a k-dominating set in 𝐺(𝑉, 𝐸) if every vertex in 𝑉not in 𝑆has atleast k neighbour in 𝑆. The k-domination number of 𝐺, denoted by 𝛾𝑘(𝐺), is the minimum cardinality of a k-dominating set of 𝐺. When 𝑘 = 2, k-dominating set is a called 2-dominating set. A dominating set Swith the additional property that the subgraph induced by S contains a perfect matching is called a paired-dominating set. The k- domination number andpaired domination numberof Gare the minimum cardinality of a k -dominating set andthe minimum cardinality of a paired dominating set respectively of G.In this paper, we obtain2- domination numberand paired domination number for special classes of graphs such as path necklace, cycle necklace and complete necklace networks.Further, we determine a lower bound for 2-domination numberof special classes of graphs.

KEYWORDS

Dominating set, 2-dominating Set, P-necklace, C- necklace, K-necklace, Star Necklace Graphs.

AMS Classification: 05C69, 05C76.

Introduction

The domination in graphs is one of the concepts in graph theory which has attracted many researchers to work on it because it has the potential to solve many real life problems involving design and analysis of communication network as well as defense surveillance. Many variants of domination models are available in the existing literature such as independent domination, total domination, connected domination, edge domination and so on. In this paper, we obtain2- domination numberand paired domination number for special classes of graphs.

For𝑣 ∈ 𝑉 (𝐺), the open neighbourhood of 𝑣, denoted as 𝑁𝐺(𝑣), is the set of vertices adjacent with 𝑣; and the closed neighbourhood of 𝑣, denoted by 𝑁𝐺[𝑣], is 𝑁𝐺(𝑣) {𝑣}. For a set 𝑆 ⊆ 𝑉 (𝐺), the open neighbourhood of 𝑆 is defined as 𝑁𝐺 𝑆 = 𝑣𝜖𝑆𝑁𝐺 𝑣 and the closed neighbourhood of 𝑆 is defined as𝑁𝐺 𝑆 = 𝑁𝐺 𝑆 𝑆 For brevity, we denote 𝑁𝐺 𝑆 by 𝑁 𝑆 and 𝑁𝐺 𝑆 by 𝑁[𝑆]. For a graph 𝐺(𝑉, 𝐸), 𝑆 ⊆ 𝑉 is a dominating set of 𝐺 if every vertex in 𝑉 \𝑆 has at least one neighbour in 𝑆.The domination number of 𝐺, denoted by γ(G), is the minimum cardinality of a dominating set of 𝐺. A set 𝑆 ⊆ 𝑉is a k-dominating set in 𝐺(𝑉, 𝐸) if every vertex in 𝑉not in 𝑆has atleast k- neighbours in 𝑆. The k-domination number of 𝐺, denoted by 𝛾𝑘(𝐺), is the minimum cardinality of a k-dominating set of 𝐺. When 𝑘 = 2, k-dominating set is a called 2-dominating set. The 2- domination numberof Gis the minimum cardinality of a 2- dominating setof G.

If no two edges in it are adjacent to G, that is, an independent edge set is a set of edges without common vertices, the set of edges in the graph G is independent. A matching G in a graph is a group of independent G edges. The matching graph number ofG denoted by𝛼(𝐺), is the maximum matching cardinality of G. A perfect matching of M is a matching of any vertex of M. Agraph G paired-dominant set is a S dominant set of G, with the additional property that the G[S] subgraph induced by S contains a perfect M match (not necessarily induced). The paired- domination number of G, denoted by γ𝑝𝑟(G), is the minimum paired-dominant cardinality set in G.

In 1985Fink and Jacobson [2]introduced k-dominating set and also shown that 𝛾𝑘 𝐺 ≥ 𝛾 𝐺 + 𝑘 − 2. Haynes, Hedetniemi, and Slater [5], as well as to theexcellent survey on k-domination in graphs by Chellali, Favaron, Hansberg, and Volkmann [3] and so on.

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Main Results

In this section, we prove that the 2-domination number and paireddomination number are equal for some special classes of graphs like P- necklace, C-necklace and K-necklace and star necklace graphs and we obtain a lower bound for 2-domination number with respect to cut-vertices.

In 2019 Anitha et.al.[7] proved thata lower bound for the domination number of a graph 𝐺 and is quoted below. With this connection we have given a lower bound for 2-domination number of a graph G.

Lemma 2.1:[7]Let 𝐺 be a graph on 𝑛-vertices with cut vertices 𝑣1, 𝑣2, . . . , 𝑣𝑚,𝑚 ≥ 2. Suppose for every cut vertex 𝑣𝑖of 𝐺, there exists components 𝐻𝑖1, 𝐻𝑖2, . . . , 𝐻𝑖𝑘, 𝐾𝑖 ≥ 1, such that𝐺[ 𝑉 𝐻𝑖𝑗 𝑣𝑖 is a complete graph on 𝑟𝑖𝑗vertices, 𝑟𝑖𝑗 ≥ 3, 1 ≤ 𝑗 ≤ 𝑘𝑖, 1 ≤ 𝑖 ≤ 𝑚. Then 𝛾(𝐺) ≥ 𝑚.

A critical subgraph of H of G is defined by the following lemma in the sense that H contains at leasttwo vertices of any 2- dominating set.

Lemma 2.2: Let 𝐺 be a graph and H as shown in Figure 1(a) be a subgraph G. Then H is a critical subgraph of 𝐺.

Proof. Let 𝑆 be a 2-dominating set of G. We claim that |S|=2. Suppose not, let us assume that |S|=1.Let 𝑉(𝐻) = {𝐾𝑡𝑖} as shown in Figure 1(a).

We note that H is connected to the rest of the graph only through the cut vertices. say 𝑣𝑖, 1 ≤ 𝑖 ≤ 𝑚. Now we have the following two cases:

1. 𝑆 ∩ 𝑉(𝐻) ≠ 𝜑.

2. 𝑆 ∩ 𝑉 𝐻 = 𝜑.

In both these cases, the𝑉(𝐻) are not 2-dominated, a contradiction.

Lemma 2.3: Let 𝐺 be a graph on 𝑛-vertices with cut vertices 𝑣1, 𝑣2, . . . , 𝑣𝑚,𝑚 ≥ 2. Suppose for every cut vertex 𝑣𝑖of 𝐺, there exists components 𝐻𝑖1, 𝐻𝑖2, . . . , 𝐻𝑖𝑘, 𝐶𝑖 ≥ 3, such that𝐺[ 𝑉 𝐻𝑖𝑗 𝑣𝑖 is a cycle on 𝑟𝑖𝑗vertices, 𝑟𝑖𝑗 ≥ 3, 1 ≤ 𝑗 ≤ 𝑛, 1 ≤ 𝑖 ≤ 𝑚. Then γ2(G) ≥ 𝑚𝑛

2. Proof. We claim that |𝑆| = 𝑚𝑛

2. Let 𝐻𝑖, 1 ≤ 𝑖 ≤ 𝑚 be the components of 𝐺\𝑣𝑖such that 𝐺[𝐻𝑖 𝑣𝑖] ≅ 𝐶𝑡𝑖. There are 𝑚 vertex disjoint copies of 𝐶𝑡𝑖in 𝐺. Let 𝑆 be the 2-dominating set which contains 𝑣𝑖𝑠 of every 𝐶𝑡𝑖. Let 𝐻 is isomorphic to cycle𝐶𝑗. Then the vertices of 𝐻 is connected to through only 𝑣𝑖, 𝑖 = 1,2, . . . , 𝑚. Let us assume the contrary that there exist a 𝐻such that 𝑉 𝐻 ∩ 𝑆 = 𝑛

2− 1.Then at least one vertex in 𝐻does not 2-dominate any member of S, a contradiction. This implies that, γ2(G) ≥ 𝑚𝑛

2. P-Necklace

Definition 4. [6] Let Pmbe a path on m vertices and let Ktii be complete graphs on tivertices, 1 ≤ i ≤ m. The graph obtained by identifying any one vertex of Ktiwith ith vertex of Pm, 1 ≤ i ≤ m, is called a P-necklace and is denoted by PN Pm; Kt1, Kt2, . . . , Ktm .

Definition 5. [6] Let 𝐶𝑚be a cycle on 𝑚 vertices and let 𝐾𝑡𝑖be complete graphs on 𝑡𝑖vertices, 1 ≤ 𝑖 ≤ 𝑚. The graph obtained by identifying any one vertex of 𝐾𝑡𝑖with ith vertex of 𝐶𝑚, 1 ≤ 𝑖 ≤ 𝑚, is called a C-necklace and is denoted by 𝐶𝑁 𝐶𝑚; 𝐾𝑡1, 𝐾𝑡2, . . . , 𝐾𝑡𝑚 . See Figure 1(a).

Lemma 2.1.1: Let 𝐺 be a C-necklace and is denoted by 𝐶𝑁 𝐶𝑚; 𝐾𝑡1, 𝐾𝑡2, . . . , 𝐾𝑡𝑚 , 𝑚 ≥ 4. Then γ2(G) ≥ 2𝑚.

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Proof. In 𝐶𝑁 𝐶𝑚; 𝐾𝑡1, 𝐾𝑡2, . . . , 𝐾𝑡𝑚 ,, there are m vertex disjointcopies of H as described in Lemma 2.1.

Therefore,γ2(G) ≥ 2𝑚.

The Algorithm given below computes the 2- domination number in P- Necklace graph PN Pm; Kt1, Kt2, . . . , Ktm .

2-Domination Algorithm in P- Necklace

Input: Let G be a P-necklace graph 𝑃𝑁 𝑃𝑚; 𝐾𝑡1, 𝐾𝑡2, . . . , 𝐾𝑡𝑚 . Algorithm: Label the vertices of 𝑃𝑁 𝑃𝑚; 𝐾𝑡1, 𝐾𝑡2, . . . , 𝐾𝑡𝑚 as follows:

(i) Label the vertices of the subgraph 𝑃𝑚of 𝐺 as 𝑣1, 𝑣2, . . . , 𝑣𝑚from left to right such that each complete graph 𝐾𝑡𝑖shares exactly one vertex of 𝑣𝑖of 𝑃𝑚, 1 ≤ 𝑖 ≤ 𝑚.

(ii) Let 𝐻𝑖, 1 ≤ 𝑖 ≤ 𝑚be the components of 𝐺\𝑣𝑖such that 𝐺[𝐻𝑖∪ 𝑣𝑖] ≅ 𝐾𝑡𝑖and selectan edge 𝑒𝑖 = 𝑢𝑖, 𝑣𝑖 , 1 ≤ 𝑖 ≤ 𝑚from each 𝐻𝑖 in 𝑆.

Output: γ2(G) = 2𝑚.

Figure 1.Red colored squared vertices constitutes the 2- dominating set of (a)C-Necklace 𝑵(𝑷𝒎, 𝑲) (b) K-Necklace 𝑵 𝑲𝒎; 𝑲𝒕𝟏, 𝑲𝒕𝟐, . . . , 𝑲𝒕𝒎

Proof of correctness: Let 𝑆be the set of 2𝑚vertices chosen from each 𝐻𝑖of 𝐺. We claim that,|𝑆| = 2𝑚. Let 𝑣1, 𝑣2, . . . , 𝑣𝑚be the vertices of the subgraph 𝑃𝑚of 𝐺such that each complete graph 𝐾𝑡𝑖shares exactly one vertex of 𝑣𝑖of 𝑃𝑚, 1 ≤ 𝑖 ≤ 𝑚. It is easy to see that 𝑣𝑖is a cut vertex of 𝐺, 1 ≤ 𝑖 ≤ 𝑚. Let 𝐻𝑖be the components of 𝐺\

𝑣𝑖such that 𝐺[𝐻𝑖∪ 𝑣𝑖] ≅ 𝐾𝑡𝑖. To prove each vertex in P-necklace is dominated, it is enough to prove that the vertices considered in 𝑆dominates the each of [𝐻𝑖∪ 𝑣𝑖] ≅ 𝐾𝑡𝑖, 1 ≤ 𝑖 ≤ 𝑚. |𝑆| = 2𝑚.Moreover, the vertices in 𝑆are pairwise adjacent. This implies thatγ2 G = γ𝑝𝑟 G = 2𝑚.

Theorem 1. Let 𝐺 be a P-necklace graph

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𝑃𝑁 𝑃𝑚; 𝐾𝑡1, 𝐾𝑡2, . . . , 𝐾𝑡𝑚 𝑜𝑟 𝑎 𝐶 − 𝑛𝑒𝑐𝑘𝑙𝑎𝑐𝑒 𝑔𝑟𝑎𝑝ℎ 𝐶𝑁 𝐶𝑚; 𝐾𝑡1, 𝐾𝑡2, . . . , 𝐾𝑡𝑚 , 𝑛 ≥ 3 , 𝑡𝑖 ≥ 3 , 1 ≤ 𝑖 ≤ 𝑚 on𝑛 = 𝑚 𝑡𝑖

𝑖=1 vertices. Then γ2(G) = 2𝑚

K- Necklace

Definition 6. [6] Let 𝐾𝑚and𝐾𝑡𝑖be complete graphs on m and 𝑡𝑖vertices respectively, 1 ≤ 𝑖 ≤ 𝑚. The graph obtained by identifying any one vertex of 𝐾𝑡𝑖with the ith vertex of𝐾𝑚,1 ≤ 𝑖 ≤ 𝑚, is called a 𝐾-necklace and is denoted by 𝐾𝑁 𝐾𝑚; 𝐾𝑡1, 𝐾𝑡2, . . . , 𝐾𝑡𝑚 . See Figure 3.

Definition 7. [6] Let 𝐾1,𝑚be a star graph on 𝑚 + 1vertices and let 𝐾𝑡𝑖be a complete graphs on 𝑡𝑖vertices, 1 ≤ 𝑖 ≤ 𝑚. The graph obtained by identifying any one vertex of 𝐾𝑡𝑖with 𝑖th vertex of 𝐾1,𝑚, 1 ≤ 𝑖 ≤ 𝑚, is called a star necklace and is denoted by 𝑁 𝐾1,𝑚; 𝐾𝑡1, 𝐾𝑡2, . . . , 𝐾𝑡𝑚 See Figure1(b).

2- Domination Algorithm in K-Necklace Graph Input: The K-necklace𝐾𝑁 𝐾𝑚; 𝐾𝑡1, 𝐾𝑡2, . . . , 𝐾𝑡𝑚 .

Algorithm: Label the vertices of𝐾𝑁 𝐾𝑚; 𝐾𝑡1, 𝐾𝑡2, . . . , 𝐾𝑡𝑚 as follows:

(i) Label the vertices of the subgraph 𝐾𝑚of 𝐺as 𝑣1, 𝑣2, . . . , 𝑣𝑚in the clock wise direction such that each complete graph 𝐾𝑡𝑖shares exactly one vertex of 𝑣𝑖of 𝐾𝑚, 1 ≤ 𝑖 ≤ 𝑚.

(ii) Let 𝐻𝑖, 1 ≤ 𝑖 ≤ 𝑚be the components of 𝐺\𝑣𝑖such that 𝐺[𝐻𝑖∪ 𝑣𝑖] ≅ 𝐾𝑡𝑖, and select 2- vertices from each 𝐻𝑖in 𝑆.

Output: γ2 G = 2𝑚

Proof of Correctness: Let 𝑆 be the set of 2𝑚vertices chosen from each 𝐻𝑖of 𝐺. We claim that, |𝑆| = 2𝑚. Let 𝑣1, 𝑣2, . . . , 𝑣𝑚be the vertices of the subgraph 𝐾𝑚of 𝐺such that each complete graph 𝐾𝑡𝑖shares exactly one vertex of 𝑣𝑖of 𝐾𝑚, 1 ≤ 𝑖 ≤ 𝑚. It is easy to see that 𝑣𝑖is a cut vertex of 𝐺, 1 ≤ 𝑖 ≤ 𝑚. Let 𝐻𝑖be the components of 𝐺\

𝑣𝑖such that 𝐺[𝐻𝑖∪ 𝑣𝑖] ≅ 𝐾𝑡𝑖. To prove each vertex in 𝐾 −necklace is dominated, it is enough to prove that the vertices considered in 𝑆dominates the each of (𝐻𝑖∪ 𝑣𝑖) ≅ 𝐾𝑡𝑖, 1 ≤ 𝑖 ≤ 𝑚. By Lemma 2, domination number of 𝐾𝑡𝑖is 2. There are mdistinct components of 𝐻 in 𝐺. Moreover, the vertices in 𝑆are pairwise adjacent. Therefore,

|𝑆| = 2𝑚. This implies that γ2 G = γ𝑝𝑟 G = 2𝑚.

Theorem 2. Let 𝐺 be the 𝐾 −necklace 𝐾𝑁 𝐾𝑚; 𝐾𝑡1, 𝐾𝑡2, . . . , 𝐾𝑡𝑚 𝑜𝑟 𝑎 𝑠𝑡𝑎𝑟necklace 𝐾𝑁 𝐾𝑚; 𝐾𝑡1, 𝐾𝑡2, . . . , 𝐾𝑡𝑚 , 𝑚 ≥ 4, 𝑡𝑖 ≥ 4, 1 ≤ 𝑖 ≤ 𝑚. Then γ2 G = γ𝑝𝑟 G = 2𝑚.

2-Domination Number of a Graph G, 𝑯𝒊 ≅ 𝑪

𝒏

In this section, we solve 2- domination number for special classes of graphs for which each 𝐻𝑖 ≅ 𝐶𝑛

Definition 6. Let Pmbe a path on m vertices and let Ctii be cycle on tivertices, 1 ≤ i ≤ n. The graph obtained by identifying any one vertex of Ctiwith ith vertex of Pm, 1 ≤ i ≤ n, is called a C-necklace and is denoted by PN Pm; Ct1, Ct2, . . . , Ctm . Let 𝐶𝑚be a cycle on 𝑚 vertices and let 𝐶𝑡𝑖be complete graphs on 𝑡𝑖vertices, 1 ≤ 𝑖 ≤ 𝑛.

The graph obtained by identifying any one vertex of 𝐶𝑡𝑖with ith vertex of 𝐶𝑚, 1 ≤ 𝑖 ≤ 𝑛, is called a C-necklace and is denoted by 𝐶𝑁 𝐶𝑚; 𝐶𝑡1, 𝐶𝑡2, . . . , 𝐶𝑡𝑚 . See Figure 2.

Definition 7. [6] Let 𝐾1,𝑚be a star graph on 𝑚 + 1vertices and let 𝐶𝑡𝑖be a cycle on 𝑡𝑖vertices, 1 ≤ 𝑖 ≤ 𝑛. The graph obtained by identifying any one vertex of 𝐶𝑡𝑖with 𝑖th vertex of 𝐾1,𝑚, 1 ≤ 𝑖 ≤ 𝑛, is called a star necklace and is denoted by 𝑁 𝐾1,𝑚; 𝐶𝑡1, 𝐶𝑡2, . . . , 𝐶𝑡𝑚 . See Figure3.

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2-Domination Algorithm in Star- Necklace

Input: Let G be a P-necklace graph by 𝑁 𝐾1,𝑚; 𝐶𝑡1, 𝐶𝑡2, . . . , 𝐶𝑡𝑚 ..

Algorithm: Label the vertices of 𝑃𝑁 𝐾1,𝑚; 𝐶𝑡1, 𝐶𝑡2, . . . , 𝐶𝑡𝑚 as follows:

(i) Label the root vertex of 𝐾1,𝑚as v0.

(ii) Label the vertices that are adjacent to root vertex v0 consecutively 𝑣1, 𝑣2, . . . , 𝑣𝑚in the clock wise direction.

(iii) Let 𝐻𝑖, 1 ≤ 𝑖 ≤ 𝑚 be the components of 𝐺\𝑣𝑖such that that 𝐺[𝐻𝑖 ∪ 𝑣𝑖] ≅ 𝐶𝑡𝑖. and select 𝑛

2vertices from each 𝐻𝑖in 𝑆.

Output: γ2(G) = 𝑚𝑛

2

Proof of Correctness: Let 𝑆 be the set of 𝑚𝑛

2vertices chosen from each 𝐻𝑖of 𝐺. We claim that, |𝑆| = 𝑚𝑛

2. Let 𝑣1, 𝑣2, . . . , 𝑣𝑚be the vertices of the subgraph 𝐾1,𝑚of 𝐺such that each cycle 𝐶𝑡𝑖 shares exactly one vertex of vi of 𝐾1,𝑚, 1 ≤ 𝑖 ≤ 𝑚. It is easy to see that 𝑣𝑖is a cut vertex of 𝐺, 1 ≤ 𝑖 ≤ 𝑚. Let 𝐻𝑖be the components of 𝐺\𝑣𝑖such that 𝐺[𝐻𝑖∪ 𝑣𝑖] ≅ 𝐾𝑡𝑖. To prove each vertex in star necklace is 2- dominated, it is enough to prove that the vertices considered in S 2-dominates the each of (𝐻𝑖 ∪ 𝑣𝑖) ≅ 𝐶𝑡𝑖, 1 ≤ 𝑖 ≤ 𝑚. Lemma 3, domination number of 𝐶𝑡𝑖is 𝑚𝑛

2.Therefore, |𝑆| = 2𝑚. This implies that γ2(G) = 𝑚𝑛

2.

Figure 3.Red colored squared vertices constitutes the 2- dominating set of Star-Necklace 𝑲𝑵(𝑲𝟏,𝟓)

Theorem 3. Let 𝐺 be the 𝐾 −necklace 𝐾𝑁 𝐾𝑚; 𝐶𝑡1, 𝐶𝑡2, . . . , 𝐶𝑡𝑚 𝑜𝑟 𝑎 𝑠𝑡𝑎𝑟necklace 𝐾𝑁 𝐾1,𝑚; 𝐶𝑡1, 𝐶𝑡2, . . . , 𝐶𝑡𝑚 , 𝑚 ≥ 4, 𝑡𝑖 ≥ 4, 1 ≤ 𝑖 ≤ 𝑚. Then γ2 G = 𝑚𝑛

2.

Theorem 4. Let 𝐺 be the 𝑃 −necklace 𝑃𝑁 𝑃𝑚; 𝐶𝑡1, 𝐶𝑡2, . . . , 𝐶𝑡𝑚 𝑜𝑟 𝑎 𝐶 −necklace 𝐶𝑁 𝐶𝑛; 𝐶𝑡1, 𝐶𝑡2, . . . , 𝐶𝑡𝑚 , 𝑚 ≥ 4, 𝑡𝑖 ≥ 4, 1 ≤ 𝑖 ≤ 𝑚. Then γ2 G = 𝑚𝑛

2.

Conclusion

In this paper, we 2-dominationnumber for P-Necklace, C-Necklace,K-Necklace and Star Necklace.Further, we obtained a lower bound for 2-domination number with respect to cut vertices. Moreover, we shown that 2-

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domination number and paired domination number are equal for P-Necklace, C-Necklace, K-Necklace and Star Necklace, where 𝑯𝒊 ≅ 𝑲𝒏.

References

[1] E.J. Cockayne and S.T. Hedetniemi, Towards a theory of domination in graphs, Networks,7 (1977) 247-261.

[2] J.F. Fink, M.S. Jacobson, n-domination in graphs, in: Y. Alavi, A.J. Schwenk (Eds.), GraphTheory with Applications to Algorithms and Computer Science,Wiley, New York, 1985, pp.283–300.

[3] J.R. Griggs, D.J. Kleitman, Independence and the Havel-Hakimi Residue, Discrete Math 127 (1994)209–

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[4] A. Hansberg, D. Meierling, L. Volkmann, Independence and k-domination in graphs, Int. J. Comput. Math., 5 (2011) 905–915.

[5] T.W. Haynes, S.T. Hedetniemi, P.J. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, Inc., New York, 1998.

[6] R.S. Rajan, N. Parthiban and T.M. Rajalaxmi, Embedding of recursive circulant into certain necklace graphs, Mathematics in Computer Science, 9(2) (2015) 253-263.

[7] J. Anitha, V. Suganya and G. Jayaraman, Independent domination number in necklacegraph,Journal of International Pharmaceutical Research 46(1) (2019) 388-392.

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