### 1973 http://annalsofrscb.ro

**2-Domination Number in Special Classes of Graphs **

**J. Anitha**

^{1*}**, G. Arul Freeda Vinodhini**

^{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 𝛾_{𝑘}(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 𝑁_{𝐺}(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 𝑁*_{𝐺} 𝑆 = _{𝑣𝜖𝑆}𝑁_{𝐺} 𝑣 and the closed neighbourhood of 𝑆 is defined as𝑁_{𝐺} 𝑆 = 𝑁𝐺 𝑆 𝑆 For brevity, we denote
𝑁_{𝐺} 𝑆 by 𝑁 𝑆 and 𝑁_{𝐺} 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 γ_{𝑝𝑟}(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.

### 1974 http://annalsofrscb.ro

**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 𝑣**_{1}, 𝑣_{2}, . . . , 𝑣_{𝑚},𝑚 ≥ 2. Suppose for every cut vertex
𝑣_{𝑖}of 𝐺, there exists components 𝐻_{𝑖1}, 𝐻_{𝑖2}, . . . , 𝐻_{𝑖𝑘}, 𝐾_{𝑖} ≥ 1, such that𝐺[ 𝑉 𝐻_{𝑖𝑗} 𝑣_{𝑖} is a complete graph on
𝑟_{𝑖𝑗}vertices, 𝑟_{𝑖𝑗} ≥ 3, 1 ≤ 𝑗 ≤ 𝑘_{𝑖}, 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) =**
{𝐾_{𝑡𝑖}} 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 𝑣**_{1}, 𝑣_{2}, . . . , 𝑣_{𝑚},𝑚 ≥ 2. Suppose for every cut vertex
𝑣_{𝑖}of 𝐺, there exists components 𝐻_{𝑖1}, 𝐻_{𝑖2}, . . . , 𝐻_{𝑖𝑘}, 𝐶_{𝑖} ≥ 3, such that𝐺[ 𝑉 𝐻_{𝑖𝑗} 𝑣_{𝑖} is a cycle on 𝑟_{𝑖𝑗}vertices, 𝑟_{𝑖𝑗} ≥
3, 1 ≤ 𝑗 ≤ 𝑛, 1 ≤ 𝑖 ≤ 𝑚. Then γ2(G) ≥ 𝑚^{𝑛}

2.
**Proof. We claim that |𝑆| = 𝑚**^{𝑛}

2. Let 𝐻𝑖, 1 ≤ 𝑖 ≤ 𝑚 be the components of 𝐺\𝑣𝑖such that 𝐺[𝐻𝑖 𝑣𝑖] ≅ 𝐶𝑡𝑖. There
are 𝑚 vertex disjoint copies of 𝐶_{𝑡𝑖}in 𝐺. Let 𝑆 be the 2-dominating set which contains 𝑣_{𝑖}𝑠 of every 𝐶_{𝑡𝑖}. Let 𝐻 is
isomorphic to cycle𝐶_{𝑗}. Then the vertices of 𝐻 is connected to through only 𝑣_{𝑖}, 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, γ_{2}(G) ≥ 𝑚^{𝑛}

2.
**P-Necklace **

**Definition 4. [6] Let P**mbe a path on m vertices and let Kt_{i}i be complete graphs on tivertices, 1 ≤ i ≤ m. The graph
obtained by identifying any one vertex of Kt_{i}with i^{th }vertex of Pm, 1 ≤ i ≤ m, is called a P-necklace and is denoted
by PN Pm; Kt_{1}, Kt_{2}, . . . , Kt_{m} .

**Definition 5. **[6] *Let 𝐶*_{𝑚}*be a cycle on 𝑚 vertices and let 𝐾*_{𝑡}_{𝑖}*be complete graphs on 𝑡*_{𝑖}*vertices, 1 ≤ 𝑖 ≤ 𝑚. The *
*graph obtained by identifying any one vertex of 𝐾*_{𝑡}_{𝑖}*with i*^{th }*vertex of 𝐶*_{𝑚}*, 1 ≤ 𝑖 ≤ 𝑚, is called a C-necklace and is *
*denoted by 𝐶𝑁 𝐶*_{𝑚}; 𝐾_{𝑡}_{1}, 𝐾_{𝑡}_{2}, . . . , 𝐾_{𝑡}_{𝑚} . See Figure 1(a).

**Lemma 2.1.1: Let 𝐺 be a C-necklace and is denoted by 𝐶𝑁 𝐶**𝑚; 𝐾𝑡_{1}, 𝐾𝑡_{2}, . . . , 𝐾𝑡_{𝑚} , 𝑚 ≥ 4. Then γ2(G) ≥ 2𝑚.

### 1975 http://annalsofrscb.ro

*Proof. * In 𝐶𝑁 𝐶_{𝑚}; 𝐾_{𝑡}_{1}, 𝐾_{𝑡}_{2}, . . . , 𝐾_{𝑡}_{𝑚} ,, 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 P_{m}; K_{t}_{1}, K_{t}_{2}, . . . , K_{t}_{m} .

**2-Domination Algorithm in P- Necklace **

**Input: Let G be a P-necklace graph 𝑃𝑁 𝑃**_{𝑚}; 𝐾_{𝑡}_{1}, 𝐾_{𝑡}_{2}, . . . , 𝐾_{𝑡}_{𝑚} .
**Algorithm: Label the vertices of 𝑃𝑁 𝑃**_{𝑚}; 𝐾_{𝑡}_{1}, 𝐾_{𝑡}_{2}, . . . , 𝐾_{𝑡}_{𝑚} as follows:

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

(ii) Let 𝐻𝑖*, *1 ≤ 𝑖 ≤ 𝑚be the components of 𝐺\𝑣𝑖such that 𝐺[𝐻𝑖∪ 𝑣𝑖] ≅ 𝐾𝑡_{𝑖}and selectan edge 𝑒𝑖 =
𝑢_{𝑖}, 𝑣_{𝑖} , 1 ≤ 𝑖 ≤ 𝑚from each 𝐻_{𝑖 }in 𝑆.

**Output: γ**_{2}(G) = 2𝑚.

**Figure 1.Red colored squared vertices constitutes the 2- dominating set of (a)C-Necklace 𝑵(𝑷**_{𝒎}, 𝑲) (b) K-Necklace
𝑵 𝑲_{𝒎}; 𝑲_{𝒕}_{𝟏}, 𝑲_{𝒕}_{𝟐}, . . . , 𝑲_{𝒕}_{𝒎}

**Proof of correctness: Let 𝑆be the set of **2𝑚vertices chosen from each 𝐻_{𝑖}of 𝐺. We claim that,|𝑆| = 2𝑚. Let
𝑣1, 𝑣2, . . . , 𝑣𝑚be the vertices of the subgraph 𝑃𝑚of 𝐺such that each complete graph 𝐾𝑡_{𝑖}shares exactly one vertex of
𝑣_{𝑖}of 𝑃_{𝑚}, 1 ≤ 𝑖 ≤ 𝑚. It is easy to see that 𝑣_{𝑖}is a cut vertex of 𝐺*, 1 ≤ 𝑖 ≤ 𝑚. Let 𝐻*_{𝑖}be the components of 𝐺\

𝑣_{𝑖}such that 𝐺[𝐻_{𝑖}∪ 𝑣_{𝑖}] ≅ 𝐾_{𝑡}_{𝑖}. To prove each vertex in *P-necklace is dominated, it is enough to prove that the *
vertices considered in 𝑆dominates the each of [𝐻_{𝑖}∪ 𝑣_{𝑖}] ≅ 𝐾_{𝑡}_{𝑖}*, 1 ≤ 𝑖 ≤ 𝑚. |𝑆| = 2𝑚.Moreover, the vertices in *
𝑆are pairwise adjacent. This implies thatγ_{2} G = γ_{𝑝𝑟} G = 2𝑚.

**Theorem ** **1. ** *Let * 𝐺 * be * *a * *P-necklace * *graph *

### 1976 http://annalsofrscb.ro

𝑃𝑁 𝑃_{𝑚}; 𝐾_{𝑡}_{1}, 𝐾_{𝑡}_{2}, . . . , 𝐾_{𝑡}_{𝑚} 𝑜𝑟 𝑎 𝐶 − 𝑛𝑒𝑐𝑘𝑙𝑎𝑐𝑒 𝑔𝑟𝑎𝑝ℎ 𝐶𝑁 𝐶_{𝑚}; 𝐾_{𝑡}_{1}, 𝐾_{𝑡}_{2}, . . . , 𝐾_{𝑡}_{𝑚} , 𝑛 ≥ 3 *, * 𝑡𝑖 ≥ 3 *, * 1 ≤ 𝑖 ≤
𝑚 on𝑛 = 𝑚 𝑡𝑖

𝑖=1 *vertices. Then γ*2(G) = 2𝑚

**K- Necklace **

**Definition 6. **[6] *Let 𝐾*𝑚*and𝐾*𝑡_{𝑖}*be complete graphs on m and *𝑡𝑖*vertices respectively, 1 ≤ 𝑖 ≤ 𝑚. The graph *
*obtained by identifying any one vertex of 𝐾*𝑡_{𝑖}*with the i*^{th }*vertex of𝐾*𝑚*,1 ≤ 𝑖 ≤ 𝑚, is called a 𝐾-necklace and is *
*denoted by 𝐾𝑁 𝐾*_{𝑚}; 𝐾_{𝑡}_{1}, 𝐾_{𝑡}_{2}, . . . , 𝐾_{𝑡}_{𝑚} . See Figure 3.

**Definition 7. [6] Let 𝐾**1,𝑚*be a star graph on 𝑚 + 1vertices and let 𝐾*𝑡_{𝑖}*be a complete graphs on 𝑡*𝑖*vertices, 1 ≤ 𝑖 ≤*
𝑚. The graph obtained by identifying any one vertex of 𝐾_{𝑡}_{𝑖}*with 𝑖*^{th }*vertex of 𝐾*_{1,𝑚}, 1 ≤ 𝑖 ≤ 𝑚, is called a star
*necklace and is denoted by 𝑁 𝐾*1,𝑚; 𝐾𝑡_{1}, 𝐾𝑡_{2}, . . . , 𝐾𝑡_{𝑚} See Figure1(b).

**2- Domination Algorithm in K-Necklace Graph **
**Input: The K-necklace𝐾𝑁 𝐾**_{𝑚}; 𝐾_{𝑡}_{1}, 𝐾_{𝑡}_{2}, . . . , 𝐾_{𝑡}_{𝑚} .

**Algorithm: Label the vertices of𝐾𝑁 𝐾**_{𝑚}; 𝐾_{𝑡}_{1}, 𝐾_{𝑡}_{2}, . . . , 𝐾_{𝑡}_{𝑚} as follows:

(i) Label the vertices of the subgraph 𝐾_{𝑚}of 𝐺as 𝑣_{1}, 𝑣_{2}, . . . , 𝑣_{𝑚}in the clock wise direction such that each
complete graph 𝐾𝑡_{𝑖}shares exactly one vertex of 𝑣𝑖of 𝐾𝑚*, 1 ≤ 𝑖 ≤ 𝑚. *

(ii) Let 𝐻𝑖*, 1 ≤ 𝑖 ≤ 𝑚be the components of 𝐺\𝑣*𝑖such that 𝐺[𝐻𝑖∪ 𝑣𝑖] ≅ 𝐾𝑡_{𝑖}, and select 2- vertices from each
𝐻_{𝑖}in 𝑆.

**Output: γ**_{2} 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 𝐾𝑡_{𝑖}shares exactly one vertex
of 𝑣_{𝑖}of 𝐾_{𝑚}, 1 ≤ 𝑖 ≤ 𝑚. It is easy to see that 𝑣_{𝑖}is a cut vertex of 𝐺, 1 ≤ 𝑖 ≤ 𝑚. Let 𝐻_{𝑖}be 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) ≅ 𝐾𝑡_{𝑖}, 1 ≤ 𝑖 ≤ 𝑚. By Lemma 2, domination number of
𝐾𝑡_{𝑖}is 2. There are *mdistinct components of 𝐻 in 𝐺. Moreover, the vertices in 𝑆are pairwise adjacent. Therefore, *

|𝑆| = 2𝑚. This implies that γ2 G = γ_{𝑝𝑟} G = 2𝑚.

**Theorem 2. ***Let *𝐺* be the *𝐾 −*necklace *𝐾𝑁 𝐾𝑚; 𝐾𝑡_{1}, 𝐾𝑡_{2}, . . . , 𝐾𝑡_{𝑚} 𝑜𝑟 𝑎 𝑠𝑡𝑎𝑟*necklace *𝐾𝑁 𝐾𝑚; 𝐾𝑡1, 𝐾𝑡2, . . . , 𝐾𝑡𝑚 *, *
𝑚 ≥ 4, 𝑡𝑖 ≥ 4, 1 ≤ 𝑖 ≤ 𝑚. Then γ_{2} G = γ_{𝑝𝑟} 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 Ct_{i}i be cycle on tivertices, 1 ≤ i ≤ n. The graph obtained by
identifying any one vertex of Ct_{i}with i^{th }vertex of Pm, 1 ≤ i ≤ n, is called a C-necklace and is denoted by
PN Pm; Ct_{1}, Ct_{2}, . . . , Ct_{m} . Let 𝐶𝑚*be a cycle on 𝑚 vertices and let 𝐶*𝑡_{𝑖}*be complete graphs on 𝑡*𝑖*vertices, 1 ≤ 𝑖 ≤ 𝑛. *

*The graph obtained by identifying any one vertex of 𝐶*𝑡_{𝑖}*with i*^{th }*vertex of 𝐶*𝑚*, 1 ≤ 𝑖 ≤ 𝑛, is called a C-necklace and *
*is denoted by 𝐶𝑁 𝐶*𝑚; 𝐶𝑡_{1}, 𝐶𝑡_{2}, . . . , 𝐶𝑡_{𝑚} . See Figure 2.

**Definition 7. **[6] *Let 𝐾*1,𝑚*be a star graph on 𝑚 + 1vertices and let 𝐶*𝑡_{𝑖}*be a cycle on 𝑡*𝑖*vertices, 1 ≤ 𝑖 ≤ 𝑛. The *
*graph obtained by identifying any one vertex of 𝐶*𝑡_{𝑖}*with 𝑖*^{th }*vertex of 𝐾*1,𝑚, 1 ≤ 𝑖 ≤ 𝑛, is called a star necklace and
*is denoted by 𝑁 𝐾*1,𝑚; 𝐶𝑡_{1}, 𝐶𝑡_{2}, . . . , 𝐶𝑡_{𝑚} . See Figure3.

### 1977 http://annalsofrscb.ro

**2-Domination Algorithm in Star- Necklace **

**Input: Let G be a P-necklace graph by 𝑁 𝐾**_{1,𝑚}; 𝐶_{𝑡}_{1}, 𝐶_{𝑡}_{2}, . . . , 𝐶_{𝑡}_{𝑚} ..

**Algorithm: Label the vertices of 𝑃𝑁 𝐾**_{1,𝑚}; 𝐶_{𝑡}_{1}, 𝐶_{𝑡}_{2}, . . . , 𝐶_{𝑡}_{𝑚} as follows:

(i) Label the root vertex of 𝐾_{1,𝑚}as v0.

(ii) Label the vertices that are adjacent to root vertex *v*_{0 }consecutively 𝑣_{1}, 𝑣_{2}, . . . , 𝑣_{𝑚}in 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 𝐻_{𝑖}in 𝑆.

**Output: γ**2(G) = 𝑚^{𝑛}

2

**Proof of Correctness: Let 𝑆 be the set of **𝑚^{𝑛}

2vertices chosen from each 𝐻_{𝑖}of 𝐺. We claim that, |𝑆| = 𝑚^{𝑛}

2.
Let 𝑣1, 𝑣2, . . . , 𝑣𝑚be the vertices of the subgraph 𝐾1,𝑚of 𝐺such that each cycle 𝐶𝑡𝑖 shares exactly one vertex of *v** _{i }*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 (𝐻*𝑖 ∪ 𝑣𝑖) ≅ 𝐶𝑡

_{𝑖}, 1 ≤ 𝑖 ≤ 𝑚. Lemma 3, domination number of 𝐶𝑡

_{𝑖}is 𝑚

^{𝑛}

2.Therefore, |𝑆| = 2𝑚. This implies that γ2(G) = 𝑚^{𝑛}

2.

**Figure 3.Red colored squared vertices constitutes the 2- dominating set of Star-Necklace 𝑲𝑵(𝑲**_{𝟏,𝟓})

**Theorem 3. ***Let *𝐺* be the *𝐾 −*necklace *𝐾𝑁 𝐾_{𝑚}; 𝐶_{𝑡}_{1}, 𝐶_{𝑡}_{2}, . . . , 𝐶_{𝑡}_{𝑚} 𝑜𝑟 𝑎 𝑠𝑡𝑎𝑟*necklace *𝐾𝑁 𝐾_{1,𝑚}; 𝐶_{𝑡}_{1}, 𝐶_{𝑡}_{2}, . . . , 𝐶_{𝑡}_{𝑚} *, *
𝑚 ≥ 4, 𝑡𝑖 ≥ 4, 1 ≤ 𝑖 ≤ 𝑚. Then γ_{2} G = 𝑚^{𝑛}

2.

**Theorem 4. ** *Let *𝐺* be the *𝑃 −*necklace *𝑃𝑁 𝑃_{𝑚}; 𝐶_{𝑡}_{1}, 𝐶_{𝑡}_{2}, . . . , 𝐶_{𝑡}_{𝑚} 𝑜𝑟 𝑎 𝐶 −*necklace *𝐶𝑁 𝐶_{𝑛}; 𝐶_{𝑡}_{1}, 𝐶_{𝑡}_{2}, . . . , 𝐶_{𝑡}_{𝑚} *, *
𝑚 ≥ 4, 𝑡𝑖 ≥ 4, 1 ≤ 𝑖 ≤ 𝑚. Then γ2 G = 𝑚^{𝑛}

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 ≅ 𝑲_{𝒏}.

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[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
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