Global 2-Point Set Domination Number of a Graph - Science Direct

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Electronic Notes in Discrete Mathematics 53 (2016) 213–224 www.elsevier.com/locate/endm

Global 2-Point Set Domination Number of a Graph Purnima Gupta 1 and Deepti Jain 2 Sri Venkateswara College University of Delhi Delhi, India.

Abstract A set D of vertices in a graph G = (V, E) is called a 2-point set dominating set of G if for every set T ⊆ V − D there exists a non-empty set S ⊆ D containing at most two vertices such that the induced subgraph S ∪ T  is connected. A set D ⊆ V (G) is called a global 2-point set dominating set of G if D is a 2-point set dominating set ¯ The global 2-point set domination number (2-point set domination of both G and G. number) is the minimum cardinality of a global 2-point set dominating set (2-point set dominating set) in G. In this paper we determine bounds on the global 2-point set domination number of a graph in terms of other graph invariants. We have also given relation between global 2-point set domination number and 2-point set domination number for some classes of graphs. Keywords: 2-Point Set Domination, Global 2-Point Set Domination.

1 2

Email: [email protected] Email: [email protected]

http://dx.doi.org/10.1016/j.endm.2016.05.019 1571-0653/© 2016 Elsevier B.V. All rights reserved.

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Introduction

By a graph G = (V, E) we mean a finite, undirected graph with neither loops nor multiple edges. The order |V | and size |E| of G are denoted by n and m respectively. For graph theoretic terminology we refer to West [7]. The open neighborhood of any vertex v in G is N (v) = {x : xv ∈ E(G)} and closed neighborhood of a vertex v in G is N [v] = N (v) ∪ {v}. The degree |N (v)| of a vertex v ∈ V (G) in the graph G is denoted by deg(v) and the maximum degree (minimum degree) in the graph G is denoted by (G) (δ(G)). For a set S ⊆ V (G) the open (closed) neighborhood N (S) (N [S]) in G is defined as N (S) = ∪v∈S N (v) (N [S] = ∪v∈S N [v]). For a separable graph G, the set of all blocks in G is denoted by B(G) and the set of all blocks at a cut-vertex w is denoted by Bw (G). A set D ⊆ V (G) in a graph G is a dominating set of G if for every vertex v in V − D, there exists a vertex u ∈ D such that v is adjacent to u. The domination number of G, denoted by γ(G), is the minimum cardinality of a dominating set of G [4]. A set D ⊆ V (G) in a graph G is a point -set dominating set (or, in short, psd-set) of G if for every subset S ⊆ V − D, there exists a vertex v ∈ D such that the subgraph S ∪ {v} is connected. The point-set domination number of G, denoted by γps (G), is the minimum cardinality of a psd-set [6]. In [3], authors studied 2-point set domination in graphs which is defined as follows: Definition 1.1 A set D ⊆ V (G) of vertices in G = (V, E) is a 2-point set dominating set (in short 2-psd set) of G if for every set S ⊆ V − D there exists a non-empty set T ⊆ D containing at most two vertices such that the subgraph S ∪ T  is connected. The minimum cardinality of a 2-psd set of G is called the 2-point set domination number of G which is denoted by γ2ps (G) and the corresponding 2-psd set is called a γ2ps -set of G or γ2ps (G)-set. The set of all 2-psd sets of G is denoted by D2ps (G). ¯ = (V¯ , E) ¯ of G = (V, E) is the graph with vertex set The complement G ¯ ¯ V = {¯ v : v ∈ V } and E = {{¯ u, v¯} : u¯, v¯ ∈ V¯ , u = v and {u, v} ∈ / E}. The ¯ is denoted by γ2ps (G). ¯ To simplify the 2-point set domination number of G ¯ The same convention notation we employ γ2ps ≡ γ2ps (G) and γ¯2ps ≡ γ2ps (G). ¯ will apply to other graphoidal invariants of G and G. A set D ⊆ V (G) is called a global dominating set of G = (V, E) if it is a ¯ [5],[1],[2]. The global domination number γg dominating set of both G and G of G is the minimum cardinality of a global dominating set of G.

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Definition 1.2 A subset D ⊆ V (G) is called a global 2-point set dominating ¯ The global set of G if D is a 2-point set dominating set of both G and G. 2-point set domination number denoted by γ2pg (G) is the minimum cardinality of a global 2-psd set of G and the global 2-psd set with cardinality γ2pg (G) is called the γ2pg -set of G or γ2pg (G)-set. The set of all global 2-psd sets of G is denoted by D2pg (G). In this paper we mainly consider bounds for global 2-psd number in terms of other known graph invariants. We also give relation between global 2-point set domination number and 2-point set domination number for some classes of graphs. It is easy to see that for a graph G, γ2pg ≥ 2 because if γ2pg = 1 then G must ¯ is disconnected. Therefore have a vertex of full degree which implies that G ¯ every 2-psd set of G must contain at least two vertices, which contradicts the assumption that γ2pg = 1. We have listed γ2pg , γ2ps and γ¯2ps number of few classes of graphs in Table 1. Here Kn is the complete graph of order n, Cn is the cycle of length n, Pn is the path of length n, Wn is the wheel (Cn−1 +K1 ) and Sn is the shell graph (Pn−1 + K1 ). Note that in all the examples listed in Table 1, γ2pg = max{γ2ps , γ¯2ps } except for the case of C5 where γ2pg (C5 ) = 3 > max{γ2ps (C5 ), γ2ps (C¯5 )} = 2. Graph Kn

No. of vertices (n)

γ2ps

γ¯2ps

γ2pg

1

n

n

n−2

3

3

2

2

3

n = 6, 7

n−3

2

n−3

n≥8

n−2

2

n−2

1

2

2

n−2

2

n−2

n = 4, 5

1

4

4

n≥6

1

3

3

n

1

3

3

n n = 3, 4

Cn

Pn

n=5

n = 2, 3 n≥4

Wn = Cn−1 + K1 Sn = Pn−1 + K1

Table 1 2-psd number and global 2-psd number for some classes of graphs

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Bounds

¯ The global 2-psd number has a direct relation with 2-psd number of G and G given by the following proposition which comes immediately by the definition. Proposition 2.1 For any graph G, max{γ2ps , γ¯2ps } ≤ γ2pg ≤ γ2ps + γ¯2ps . A general class of graphs for which γ2pg = max{γ2ps , γ¯2ps } is given in the following theorem which comes immediately by the definition. ¯ is disconnected, γ2pg = max{γ2ps , γ¯2ps }. Theorem 2.2 If either G or G

2.1

Specified diameter.

In this section, we consider bounds on the global domination number of graphs with specified diameter. Theorem 2.3 Let G be any graph. (i) If diam(G) ≥ 3, then γ2pg ≤ γ2ps + 2, (ii) If diam(G) ≥ 4, then γ2pg (G) = γ2ps (G). Further, in this case every γ2ps (G)-set is a γ2pg (G)-set. Proof. (i) Let G be a graph with diam(G) = k ≥ 3 and let vertices x and y of G be such that d(x, y) = k. Then N (x) ∩ N (y) = φ, which implies that every ¯ − {x, y} is either adjacent to x or y or both in G. ¯ Therefore, vertex in V (G) ¯ Thus for any D ∈ D2ps (G), D ∪ {x, y} the set {x, y} forms a 2-psd set of G. is a global 2-psd set of G and hence γ2pg ≤ γ2ps + 2. (ii) Let G be a graph with diam(G) = k ≥ 4 and let vertices x and y of G be such that d(x, y) = k. Let D be any 2-psd set of G. Since d(u, v) ≤ 3 for any two vertices u, v ∈ V −D, therefore, at least one vertex of x and y must belong to D. Without loss of generality, assume that x ∈ D. Also, N (x) ∩ N (y) = φ implies that D must contain one vertex of N [y]. Let z ∈ N [y] ∩ D. Then d(x, z) ≥ 3. Since any two vertices at distance greater than or equal to three ¯ therefore, D is a 2-psd set of G ¯ also and hence, forms a 2-psd set of G, γ2pg (G) = γ2ps (G) and every γ2ps (G)-set is a γ2pg (G)-set. 2 Remark 2.4 The upper bound of γ2pg (G) for the graphs with diam(G) = 3 given by Theorem 2.3 is sharp. For an example consider the graph G shown in Fig 1. This is a self-complementary graph with diameter 3. For this graph ¯ γ2ps = γ¯2ps = 2. But no γ2ps (G)-set has a vertex common with any γ2ps (G)-set. Thus, γ2pg (G) = 4 = γ2ps (G) + 2.

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Fig. 1. A graph G with diameter 3 satisfying γ2pg (G) = γ2ps (G) + 2.

Remark 2.5 If γ2pg = γ2ps for a graph G, then in general every γ2ps (G)-set need not be a γ2pg (G)-set. We illustrate it with the graph shown in Fig 2. x

x

y

a

b w

z G

y

a

b w

z G

¯ Fig. 2. The set {x, y} of circled vertices forms a γ2ps -set of G but not of G.

For the graph G, γ2ps = γ2pg = 2. The set {a, b} is a γ2ps (G)-set as well as ¯ γ2pg (G)-set. However, the set {x, y} is a γ2ps (G)-set but not a 2-psd set of G and therefore, not a global 2-psd set. Remark 2.6 In Theorem 2.3, we observed that for graphs with diameter 3, γ2pg − γ2ps ≤ 2. However for graphs with diameter 2, the difference between γ2ps and γ2pg can be as large as possible. For example, for the complete bipartite graph Km,n , γ2ps (Km,n ) = 2, whereas γ2pg (Km,n ) = min{m, n} + 1 for m, n ≥ 2. 2.2

Some more upper bounds

¯ are connected. A graph G is said to be co-connected graph if both G and G Theorem 2.7 Let G be a co-connected graph with |V (G)| = n ≥ 4. Then γ2pg ≤ n − 2. Proof. Let G be a co-connected graph with |V (G)| ≥ 4. Let u, v be two adjacent vertices with degree greater than or equal to two. (Such a pair always exists if |V (G)| ≥ 4). Claim: D = V (G) − {u, v} is a global 2-psd set of G. Clearly, the set D is a 2-psd set of G. If there exists a vertex x ∈ V (G) such ¯ Thus, D ∈ D2pg (G). that x ∈ / N (u) ∪ N (v) in G, then x ∈ N (u) ∩ N (v) in G. Therefore, we assume that each vertex x ∈ V (G) belongs to N (u) ∪ N (v).

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Claim: There exist vertices y ∈ N (u) − N (v) and z ∈ N (v) − N (u) such that yz ∈ / E(G). Since G is co-connected, N (u) − N (v) = φ = N (v) − N (u). Suppose the claim is not true, i.e., for each y ∈ N (u) − N (v) and for each z ∈ N (v) − N (u), ¯ yz ∈ E(G). We shall show that there does not exist any y − z path in G. ¯ Now each ui is either adjacent Suppose y, u1 , u2 , u3 , ..., un , z is a y−z path in G. to u or to v or to both in G. If u1 v ∈ E(G) then by our assumption, u1 y ∈ ¯ a contradiction. Therefore, u1 u ∈ E(G). E(G) which implies that u1 y ∈ / E(G), ¯ Next if u2 v ∈ E(G) This implies that u1 z ∈ E(G) and hence u1 z ∈ / E(G). ¯ a contradiction. Therefore, then u1 u2 ∈ E(G). This implies that u1 u2 ∈ / E(G), ¯ Continuing u2 u ∈ E(G) which implies u2 z ∈ E(G) and hence u2 z ∈ / E(G). ¯ which implies like this we will never get a vertex ui such that ui z ∈ E(G), ¯ that G is disconnected, a contradiction. Hence the claim. Therefore, there exist vertices y ∈ N (u) − N (v) and z ∈ N (v) − N (u) such ¯ and yz ∈ E(G). ¯ This that yz ∈ / E(G). Therefore, u ∈ N (z), v ∈ N (y) in G ¯ also and hence γ2pg ≤ n − 2.2 proves that V (G) − {u, v} is a 2-psd set for G The upper bound on γ2pg given by Theorem 2.7 is attained by Cn , n ≥ 8. For any graph G we observe that for any vertex v ∈ V (G), the set V (G) − N (v) ∈ D2ps (G). Also for any two adjacent vertices u, v ∈ G, the set V (G) − [N ({u, v}) − {u, v}] ∈ D2ps (G). Therefore, we define a parameter r as follows: Definition 2.8 For a graph G, we define r = max |N(S) - S|, where maximum is taken over all subsets S of V such that |S| ≤ 2 and the subgraph S is connected. From the above definition we note the following: Note 1 For any graph G of order n, γ2ps (G) ≤ n − r. Now we give upper bounds on γ2pg (G) in terms of r,  and δ. Theorem 2.9 For any graph G, if γ2pg > γ2ps , then (i) γ2pg ≤ n − r + , where  is the maximum degree of G. (ii) γ2pg ≤ n − (r − δ) + 1, where δ is the minimum degree of G. Proof. (i) Let D be a γ2ps (G)-set and v ∈ D be arbitrary. Let X = (V − ¯ as every D) ∩ N (v). Notice that D = D ∪ X is a 2-psd set of both G and G  ¯ ¯ subset of V − D in G is contained in N (v) in G. The set X has at most  number of vertices, therefore, γ2pg ≤ |D | = |D ∪ X| ≤ γ2ps +  and by using

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Note 1, we get γ2pg ≤ n − r + . (ii)Let x ∈ V be a vertex of degree δ. Then D = {x} ∪ N (x) forms a 2-psd ¯ Therefore, for any γ2ps (G)-set D of G, D ∪ D is a global 2-psd set set of G. of G. Hence, by using Note 1, we get γ2pg (G) ≤ n − (r − δ) + 1. 2 Now consider a graph G with a pendant vertex say x and let N (x) = {u}. Then each γ2ps (G)-set must contain either x or u. Also the set {x, u} is a ¯ Thus, γ2ps (G) ≤ γ2pg (G) ≤ γ2ps (G) + 1 for a graph with a 2-psd set of G. pendant vertex. Also for a γ2ps (G)-set D of a graph G, if there exists a vertex v ∈ V − D such that N (v) ⊆ D, then D ∪ {v} is a global 2-psd set of G. Thus, γ2ps (G) ≤ γ2pg (G) ≤ γ2ps (G) + 1. 2.3

γ2pg number of some classes of graphs

Theorem 2.10 If G is a tree, then ⎧ ⎨ γ + 1, if d = 2 2ps γ2pg = ⎩ γ , otherwise. 2ps Proof. If diam(G) = 2, then G is a star and γ2ps = 1 and γ2pg = 2. Now let diam(G) ≥ 3. If  = 2, then G is a path and for Pn (n ≥ 3), γ2pg = γ2ps . So we assume that  ≥ 3. Also since V (G) − N (x) is a 2-psd set of G for any x ∈ V (G), therefore, γ2ps (G) ≤ n −  ≤ n − 3. This implies that |V − D| ≥ 3 for any 2-psd set D of G. Let D be a γ2ps (G)-set. Firstly we shall show that V − D is independent. On the contrary assume that x, y ∈ V − D be adjacent. Let z ∈ (V − D) − {x, y}. Since D is a 2-psd set of G, therefore, there exists a x − z path P1 not containing y and a y − z path P2 not containing x in G. Then P1 ∪ P2 ∪ xy contains a cycle, a contradiction to the fact that G is a tree. Thus our supposition is wrong and V − D is independent. Now two cases arise: Case(i) There exist two adjacent vertices u, v ∈ D such that V − D ⊆ N (u) ∪ N (v) in G. Since G is a tree, therefore, N (u)∩N (v)∩(V −D) = φ in G. Also since, V −D is ¯ and V −D ⊆ N (u)∪N (v) independent in G, therefore, V −D is complete in G ¯ ¯ in G. This implies that D is a 2-psd set of G also. Hence, γ2pg = γ2ps . Case(ii) There exists u ∈ D such that V − D ⊆ N (u). Since diam(G) ≥ 3, there exists a vertex v ∈ D such that d(u, v) = 2. Let x ∈ N (v)∩(V −D). Then D = (D−{v})∪{x} is also a γ2ps -set of G. Thus

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D contains two vertices u, x such that ux ∈ E(G) and V − D ⊆ N (u) ∪ N (x) in G. Hence by case (i), γ2pg = γ2ps . 2 Next we will show that for a separable graph G, γ2pg (G) and γ2ps (G) differ by at most one. Theorem 2.11 If G is a co-connected separable graph, then γ2ps (G) ≤ γ2pg (G) ≤ γ2ps (G) + 1. Proof. Let G be a separable graph and D ∈ γ2ps (G). We know that γ2ps ≤ γ2pg . We shall show that γ2pg ≤ γ2ps + 1. We have three different cases: Case (i) V − D  V (B) for some block B of G. Let w be a cut-vertex of G in B and let x ∈ V (B  ) for some B  ∈ Bw (G), B  = B. Then all vertices of V − D, except possibly w, are adja¯ Thus D ∪ {w} is a 2-psd set of G ¯ and hence a global 2-psd set cent to x in G. of G. Therefore, γ2pg ≤ γ2ps + 1. Case (ii): V − D = V (B) for some block B of G. Let v ∈ V (B) and d ∈ D be such that vd ∈ E(G). Since B is a block, d is not adjacent to any other vertex of V (B) in G and therefore, adjacent to all ¯ This shows that D ∪ {v} is a 2-psd set of G ¯ the vertices of V (B) − {v} in G. and hence γ2pg ≤ γ2ps + 1. Case (iii) V − D contains vertices of different blocks of G. Subcase (i) All the blocks having non-empty intersection with V − D are at a single cut vertex, say w. Let all vertices of V − D are adjacent to w. Since G is co-connected, there exists a vertex u ∈ D − {w} in a block B of G, which is not adjacent to w. Let v be a vertex in a block B1 of G different from B. Then D ∪ {v} is a 2-psd ¯ as all vertices of (V − D) − V (B) are adjacent to u and all vertices set for G ¯ Therefore, γ2pg ≤ γ2ps + 1. of (V − D) ∩ V (B) are adjacent to v in G. Let all vertices of V − D are not adjacent to w. In this case there exists a block B ∈ Bw (G) such that all vertices of (V − D) − N (w)  V (B). Also V (B) ∩ D ∩ N (w) = φ. Let d ∈ V (B) ∩ D ∩ N (w) and u ∈ (V − D) − V (B). ¯ as all the vertices of (V − D) − V (B) The set D ∪ {u} is a 2-psd set of G are adjacent to d and all vertices of (V − D) ∩ V (B) are adjacent to u in ¯ and du ∈ E(G). ¯ Thus, D ∪ {u} is a global 2-psd set of G and therefore, G γ2pg ≤ γ2ps + 1. Subcase (ii) All the blocks having non-empty intersection with V − D are not at a common cut-vertex. Since V − D is not contained in blocks at a single cut-vertex, therefore, there exist two vertices x1 , x2 ∈ V − D such that for every block B1 containing

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x1 and for every block B2 containing x2 , V (B1 ) ∩ V (B2 ) = φ. Consider the set {x1 , x2 } ⊆ V − D. Since D is a 2-psd set of G, therefore, there exists a set W ⊆ D, |W | ≤ 2 such that {x1 , x2 } ∪ W  is connected. Also from our choice of vertices x1 and x2 , |W | = 1, otherwise there exist blocks B1 and B2 containing x1 and x2 respectively and having a common cut-vertex. Let W = {w1 , w2 }. Then (x1 , w1 , w2 , x2 ) is a path in G. Since x1 and x2 belong to two different blocks with no common cut-vertex, therefore, x1 − x2 path must pass through cut vertices. Thus, w1 and w2 are cut vertices and are adjacent. Also all other vertices of V − D are adjacent to w1 or w2 , otherwise, we will either obtain a cycle containing x1 and x2 or there will exist blocks B1 and B2 containing x1 and x2 respectively such that V (B1 ) ∩ V (B2 ) = φ, which are contrary to our choice of x1 and x2 . Thus V − D ⊆ N (w1 ) ∪ N (w2 ). Let u ∈ V − D be such that u ∈ N (w1 ) − N (w2 ). Then the set D ∪ {u} ¯ Thus, D ∪ {u} is a global 2-psd set of G and therefore, is a 2-psd set of G. γ2pg ≤ γ2ps + 1. 2 Now we find the global 2-psd number of few classes of graphs defined as follows: First, the union G = G1 ∪ G2 has V (G) = V (G1 ) ∪ V (G2 ) and E(G) = E(G1 ) ∪ E(G2 ). For disjoint graphs G1 and G2 , the join G = G1 + G2 has V (G) = V (G1 )∪V (G2 ) and E(G) = E(G1 )∪E(G2 )∪{uv : u ∈ V (G1 ) and v ∈ V (G2 )}. A set of pairwise independent edges of G is called a matching. If M is a matching in a graph G with the property that every vertex of G is incident with an edge of M , then M is a perfect matching in G. For any graph G, we obtain a graph G∗ from the disjoint union of G and ¯ by adding the edges of a perfect matching between the corits complement G ¯ Fig 3 illustrates an example responding vertices of G and its complement G. of a graph G∗ , where G = C5 .

Fig. 3. The graph G∗ for G = C5

¯ within G∗ we call them G and G ¯ respectively. Also, To distinguish G and G ¯ be let vertices of G be denoted by {ui : i = 1, 2, ..., |V (G)|} and vertices of G denoted by {u¯j : j = 1, 2, ..., |V (G)|} where i = j represents corresponding ¯ Note that the complement of G∗ is the join G + G ¯ vertices within G and G.

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¯ minus a perfect matching between the corresponding vertices of G and G. Theorem 2.12 For any graph G = (V, E), γ2ps (G∗ ) ≤ γ2pg (G∗ ) ≤ γ2ps (G∗ ) + 1. Proof. We know that γ2ps (G∗ ) ≤ γ2pg (G∗ ). We shall show that γ2pg (G∗ ) ≤ γ2ps (G∗ ) + 1. Let D be a γ2ps (G∗ )−set and x ∈ D. The set D may contain ¯ Without loss of generality we assume that u ∈ vertices of both G and G. D ∩ V (G). We shall show that the set {u, u¯} is a 2-psd set of G¯∗ . It is sufficient to verify it for independent subsets of V (G¯∗ ) − {u, u¯} only. Let S be an independent subset of V (G¯∗ ) − {u, u¯}. The set S may contain ¯ both. We have three different cases: vertices of G and G Case (i) Let S contain vertices of G only. In this case S ∪ {¯ u} is connected. ¯ only. Case (ii) Let S contain vertices of G In this case S ∪ {u} is connected. ¯ both. Case (iii) Let S contain vertices of G and G In this case S contains only two vertices, viz, a vertex of G and the cor¯ as each vertex x ∈ V (G) is adjacent to every vertex responding vertex of G, ¯ except x¯ ∈ V (G). ¯ Let S = {x, x¯}. If ux ∈ E(G), then S ∪ {u} of V (G) ¯ and therefore, S ∪ {¯ is connected and if ux ∈ / E(G), then u¯x¯ ∈ E(G) u} is connected. This proves that {u, u¯} ∈ D2ps (G¯∗ ), where u ∈ D and hence D ∪ {¯ u} ∈ ∗ ∗ ∗ D2pg (G ). Thus γ2pg (G ) ≤ γ2ps (G ) + 1. 2 For a given graph G, the graph G+ is obtained by joining exactly one pendant vertex to each vertex of G. Theorem 2.13 Let G be a connected graph with n = |V (G)| ≥ 4. Then γ2ps (G+ ) = γ2pg (G+ ). Proof. Let G be a connected graph with n ≥ 4. If G  Kn for any n ≥ 4, then diam(G+ ) ≥ 4. Therefore, by Theorem 2.3, γ2pg (G+ ) = γ2ps (G+ ). Now assume that G ∼ = Kn for some n ≥ 4. Let Ve denote the set of pendant vertices of G+ . Let u, v ∈ V (G) and u , v  ∈ Ve be the corresponding vertices. Then D = {u, v} ∪ (Ve − {u , v  }) is a γ2ps (G+ )-set. Since |V (G)| ≥ 4, D must contain at least two pendant vertices. The distance between any two vertices of Ve is three and any two vertices of distance three in a graph forms a 2-psd set of the complement of the graph. Therefore, D is a global 2-psd set of G and hence γ2pg (G+ ) = γ2ps (G+ ). 2

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For any graph G, we obtain a graph G∗ from the disjoint union of G and G by adding the edges of a perfect matching between the corresponding vertices of the two copies of G. To help distinguish the copies of G within G∗ , we define them as G1 and G2 . Also let vertices of G1 be denoted by {ui : i = 1, 2, ..., n} and vertices of G2 be denoted by {vj : j = 1, 2, ..., n}, where i = j represents corresponding vertices within G1 and G2 . Note that the complement of G∗ is ¯+G ¯ minus a perfect matching between the corresponding vertices the join G ¯ ¯ of G and G. Theorem 2.14 For any connected graph G, γ2ps (G∗ ) ≤ γ2pg (G∗ ) ≤ γ2ps (G∗ ) + 1. Proof. For any γ2ps (G∗ )-set D, the following three cases arise: Case(i) Either D ∩ V (G1 ) = φ or D ∩ V (G2 ) = φ. Without loss of generality we assume that D ∩ V (G1 ) = φ. Since D is a 2-psd set of G, D = V (G2 ). In that case D ∪ {ui } for any i, 1 ≤ i ≤ n, is a ¯ Therefore, γ2pg ≤ γ2ps (G∗ ) + 1. 2-psd set of G. Case(ii) Either |D ∩ V (G1 )| ≥ 2 or |D ∩ V (G2 )| ≥ 2. Let |D ∩ V (G1 )| ≥ 2 and ui , uj ∈ D ∩ V (G1 ). Then {ui , uj , vi } is a 2-psd ¯ Thus, γ2pg (G∗ ) ≤ γ2ps (G∗ ) + 1. The case |D ∩ V (G2 )| ≥ 2 is also set of G. similar. Case(iii) If |D ∩ V (G1 )| = 1 and |D ∩ V (G2 )| = 1. Let ui ∈ D ∩ V (G1 ) and vj ∈ D ∩ V (G2 ), i = j. Then {ui , vi , vj } is a 2-psd ¯ If i = j, then {ui , vi , vk } for any k, 1 ≤ k ≤ n, k = i is a 2-psd set of set of G. ¯ Thus, γ2pg (G∗ ) ≤ γ2ps (G∗ ) + 1. G. 2

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