Paired-domination number of a graph and its complement
Discrete Mathematics 308 (2008) 6601–6605 www.elsevier.com/locate/disc
Note
Paired-domination number of a graph and its complement O. Favaron a,b , H. Karami c , S.M. Sheikholeslami d,∗ a Univ Paris-Sud, LRI, UMR 8623, Orsay, F-91405, France b CNRS, Orsay, F-91405, France c Department of Mathematics, Sharif University of Technology, P.O. Box 11365-9415, Tehran, Islamic Republic of Iran d Department of Mathematics, Azarbaijan University of Tarbiat Moallem, Tabriz, Islamic Republic of Iran
Received 18 July 2007; received in revised form 15 November 2007; accepted 18 November 2007 Available online 26 December 2007
Abstract A paired-dominating set of a graph G = (V, E) with no isolated vertex is a dominating set of vertices inducing a graph with a perfect matching. The paired-domination number of G, denoted by γ pr (G), is the minimum cardinality of a paired-dominating set of G. We consider graphs of order n ≥ 6, minimum degree δ such that G and G do not have an isolated vertex and we prove that – if γ pr (G) > 4 and γ pr (G) > 4, then γ pr (G) + γ pr (G) ≤ 3 + min{δ(G), δ(G)}. 2n – if δ(G) ≥ 2 and δ(G) ≥ 2, then γ pr (G) + γ pr (G) ≤ 2n 3 + 4 and γ pr (G) + γ pr (G) ≤ 3 + 2 if moreover n ≥ 21. c 2007 Elsevier B.V. All rights reserved.
Keywords: Paired-domination number; Nordhaus–Gaddum inequalities
1. Introduction We use [2,9] for the terminology and notation which are not defined here. Let G = (V (G), E(G)) be a graph of order n with no isolated vertex. The (closed) neighborhood of a vertex u is denoted by N G (u) (N G [u]) and its degree |N G (u)| by dG (u) (briefly N (u) (N [u]) and d(u) when no ambiguity on the graph is possible). A dominating set of a graph G is a set S of vertices such that every vertex of V \ S is adjacent to some vertex in S. The domination number of G, denoted by γ (G), is the minimum cardinality of a dominating set. A paired-dominating set (PDS) of G is a dominating set S such that the subgraph G[S] induced by S contains a perfect matching (not necessarily induced). Every graph without isolated vertices has a paired-dominating set since the end-vertices of any maximal matching form such a set. The paired-domination number of G, denoted by γ pr (G), is the minimum cardinality of a PDS. When G is not connected, let G 1 , . . . , G k be its components. Then γ pr (G) exists if each G i has order at least 2, and Pk γ pr (G) = i=1 γ pr (G i ). A paired-dominating set of cardinality γ pr (G) is called a γ pr (G)-set. Paired-domination number was introduced by Haynes and Slater [4,3] as a model for assigning backups to guards for security purposes. They observed in [3] that γ pr (G) is even and γ pr (G) ≤ 2γ (G) for every graph with δ(G) ≥ 1. The complement G of a graph G has vertex set V (G) and x y ∈ E(G) if and only if x y 6∈ E(G). For any graph parameter µ, bounds on ∗ Corresponding author.
E-mail addresses: [email protected] (O. Favaron), [email protected] (S.M. Sheikholeslami). c 2007 Elsevier B.V. All rights reserved. 0012-365X/$ - see front matter doi:10.1016/j.disc.2007.11.034
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Fig. 1. Graphs whose paired-domination is 2n 3 .
µ(G) + µ(G) and on µ(G)µ(G) are called Nordhaus–Gaddum inequalities. Many Nordhaus–Gaddum bounds have been obtained on various domination parameters. Some of them can be improved when additional constraints on G and G are added. For instance it has been proved Theorem A. 1. γ (G) + γ (G) ≤ n + 1 for every graph G [8]; 2. γ (G) + γ (G) ≤ n2 + 2 if δ(G), δ(G) ≥ 1 [7]; 3. γ (G) + γ (G) ≤ 2n 5 + 3 if δ(G), δ(G) ≥ 2 with some small exceptions [1]; 3n 4. γ (G) + γ (G) ≤ 8 + 2 if δ(G), δ(G) ≥ 3 with some small exceptions [1]. In what concerns paired-domination, we must suppose δ(G), δ(G) ≥ 1. From Theorem A(2) and the fact that γ pr (G) ≤ 2γ (G), Haynes and Slater immediately obtained the bound γ pr (G) + γ pr (G) ≤ 2(b n2 c + 2) for every graph G with δ(G), δ(G) ≥ 1 [3]. The same argument applied to Theorem A(3) shows that, except for some small graphs, γ pr (G) + γ pr (G) ≤ 2(b 2n 5 c + 3) for every graph G with δ(G), δ(G) ≥ 2. Our purpose in this paper is to improve this bound. We will use the following known results on γ (G) and γ pr (G). Theorem B (Hellwig and Volkmann [6]). Let G be graph of order n and diameter 2. Then γ (G) ≤ b n4 c + 1. Theorem C (Haynes and Slater [3]). For any graph G, γ pr (G) ≥ 4 if and only if diam(G) = 2. Theorem D (Haynes and Slater [3]). If G is a connected graph of order n ≥ 6 with δ(G) ≥ 2, then γ pr (G) ≤
2n 3 .
Henning improved Theorem D by determining the extremal or nearly extremal graphs. Theorem E (Henning [5]). Let G be a connected graph of order n ≥ 6 with δ(G) ≥ 2. 2n−2 1. γ pr (G) = 2n for 3 if and only if G is C 6 , C 9 or one of the graphs illustrated in Fig. 1. Hence γ pr (G) ≤ 3 n ≥ 10. 2. Moreover if n ≥ 14 then γ pr (G) = 2n−2 3 if and only if G belongs to a well-determined family H of graphs of diameter greater than 2.
We note that F1 = B1 and γ pr (B1 ) = γ pr (F1 ) = 4, and that if G ∈ {C6 , C9 , D5,5 } ∪ {Fi |2 ≤ i ≤ 6} then diam(G) > 2. An immediate consequence follows from Theorem E(1) and Theorem C. Corollary 1. Let G be a connected graph of order n ≥ 6 with δ(G), δ(G) ≥ 2 and γ pr (G) = 2n then γ pr (G) + γ pr (G) = 2n 3 + 2 and if G ∈ {B1 , F1 } then γ pr (G) + γ pr (G) = 3 + 4.
2n 3 .
If G 6∈ {B1 , F1 }
2. A bound on the sum γ pr (G) + γ pr (G) 2n 3
In this section we prove that if G is a simple graph of order n ≥ 6 with δ(G), δ(G) ≥ 2, then γ pr (G) + γ pr (G) ≤ + 4. If moreover n ≥ 21, then γ pr (G) + γ pr (G) ≤ 2n 3 + 2.
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Lemma 2. For any graph G of order n ≥ 6 with γ pr (G), γ pr (G) > 4, γ pr (G) + γ pr (G) ≤ min{δ(G), δ(G)} + 3. Proof. Since the paired-domination number is always even, γ pr (G), γ pr (G) ≥ 6. We consider a vertex x of degree δ in G and X = V \ N [x]. If N (x) ∩ N (y) = ∅ for some y ∈ X then in G, x dominates X , y dominates N (x), and {x, y} is a paired-dominating set of G, a contradiction. Therefore, N (x) ∩ N (y) 6= ∅ for all y ∈ X. Hence, N (x) dominates X but no vertex of N (x) dominates X for otherwise γ pr (G) = 2. The first claim is an immediate consequence of γ pr (G) ≥ 6. Claim 1. Let A be a subset of N (x) dominating X and B a maximal subset of A not dominating X . Then |A| ≥ 3 and |B| ≥ 2. Let S be a maximum subset of N (x) which does not dominate all vertices in X and T = N (x) \ S. By the maximality of S, every vertex t of T dominates X \ N (S) and by Claim 1, |S| ≥ 2. Claim 2. T dominates X . Proof of Claim 2. Let, to the contrary, y ∈ X be not dominated by T . By the definition of S, there exists a vertex z ∈ X not dominated by S. Since y ∈ N (S) and z ∈ X − N (S), y 6= z. Now {x, z, y, w} is a paired-dominating set of G for each w ∈ T , a contradiction. Claim 3. T has no isolated vertex. Proof of Claim 3. Let, to the contrary, y be an isolated vertex in T . By Claim 1, y does not dominate X . Let z ∈ X be not dominated by y and w ∈ X \ N (S). Obviously z ∈ N (S) and thus w 6= z. Now {x, w, y, z} is a paired-dominating set of G, a contradiction. Let M = {x1 y1 , x2 y2 , . . . , xs ys } be a maximal matching of G[T ]. Then s ≥ 1 by Claim 3 and U = T \ V (M) is an independent set of G. Claim 4. If V (M) does not dominate X , then V (M) ∪ {y} dominates X for all y ∈ T \ V (M). Proof of Claim 4. If V (M) does not dominate X , then U = T \ V (M) 6= ∅ by Claim 2. Suppose V (M) ∪ {y} does not dominate X for some y ∈ U and let w1 ∈ X be not dominated by V (M) ∪ {y}. Let w2 ∈ X \ N (S). In G, w1 dominates V (M), y dominates U since U is independent in G, and {x, w2 , y, w1 } is a paired-dominating set of G, a contradiction. Let M 0 be a maximal submatching of M such that V (M 0 ) does not dominate X and let W be a maximum subset of T containing V (M 0 ) and not dominating X . By Claims 1 and 2 and the definition of S, M 0 6= ∅, W 6= T and |S| ≥ |W |. However it is possible that M 0 = M, in which case W = V (M) by Claim 4, or that V (M 0 ) ⊆ W 6= V (M). Claim 5. If V (M 0 ) ∪ {y} dominates X for some y ∈ N (x) \ V (M 0 ), in particular if M 0 = M or if W = V (M 0 ), then γ pr (G) ≤ |V (M 0 )| + 2 and |V (M 0 )| ≥ 4. Proof of Claim 5. If V (M 0 ) ∪ {y} dominates X , then V (M 0 ) ∪ {x, y} is a paired-dominating set of G, which proves the claim since γ pr (G) ≥ 6. Claim 6. In any case, γ pr (G) ≤ |V (M 0 )| + 4. Proof of Claim 6. If M 0 = M then γ pr (G) ≤ |V (M 0 )| + 2 by Claim 5. Otherwise, let xs ys ∈ M − M 0 . By the maximality of M 0 , V (M 0 ) ∪ {xs , ys } dominates X and V (M 0 ) ∪ {xs , ys , x, a}, where a is any vertex of S, is a paireddominating set of G.
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Let T1 = T \ W . If T1 dominates X , let S1 be a maximal subset of T1 not dominating X and T2 = T1 \ S1 . If T2 dominates X , let S2 be a maximal subset of T2 not dominating X and T3 = T2 \ S2 . We continue the process until obtention of a subset Tk of T not dominating X . Thus we construct a finite chain T = T0 % T1 % · · · % Tk (k ≥ 1) such that (a) For 0 ≤ i ≤ k − 1, Ti dominates X . (b) For 0 ≤ i ≤ k − 1, Si = Ti \ Ti+1 does not dominate X , but Si ∪ {y} dominates X for each y ∈ Ti+1 . (c) Tk does not dominate X . Then S0 = W and if k ≥ 2, |Si | ≥ 2 for 1 ≤ i ≤ k − 1 by Claim 1. For all i between 0 and k − 1, let xi be a vertex of X not dominated by Si and let yi ∈ Si . Let xk be a vertex of X not dominated by Tk and yk ∈ Tk . By (b) the vertex xi is adjacent to every vertex in Ti+1 . Hence all the vertices xi are distinct. Similarly, all the vertices yi are distinct since S0 , S1 ,..., Sk−1 , Tk are disjoint. Finally let y ∈ X \ N (S). Then {x0 , y0 , x1 , y1 , . . . , xk , yk , x, y} is a paired-dominating set of G and thus γ pr (G) ≤ 2k + 4.
(∗)
On the other hand, δ(G) = |S|+|W |+ and by (∗),
Pk−1 i=1
|Ti \Ti+1 |+|Tk | ≥ 2|W |+2(k −1)+1. Therefore 2k ≤ δ(G)−2|W |+1
γ pr (G) ≤ δ(G) − 2|W | + 5.
(∗∗)
If |W | ≥ |V (M 0 )| + 2 then γ pr (G) ≤ δ(G) − 2|V (M 0 )| + 1 and by Claim 6, γ pr (G) + γ pr (G) ≤ δ(G) − |V (M 0 )| + 5 ≤ δ(G) + 3. If |W | = |V (M 0 )| + 1 and V (M 0 ) ∪ {y} dominates X for some y ∈ T \ V (M 0 ), or if W = V (M 0 ), then by Claim 5, γ pr (G) ≤ |V (M 0 )| + 2 ≤ |W | + 2 and |W | ≥ 4. Therefore by (∗∗), γ pr (G) + γ pr (G) ≤ δ(G) − |W | + 7 ≤ δ(G) + 3. If |W | = |V (M 0 )| + 1 and V (M 0 ) ∪ {y} does not dominate X for any y ∈ T \ V (M 0 ), let w1 w2 ∈ M \ M 0 . By the definition of M 0 , V (M 0 ) ∪ {w1 , w2 } dominates X and without loss of generality we may assume that W = V (M 0 ) ∪ {w1 }. If V (M 0 ) ∪ {w1 , w2 } dominates N (x), then γ pr (G) ≤ |V (M 0 )| + 2 and, as before, we have γ pr (G) + γ pr (G) ≤ δ(G) + 3. Finally suppose there exists a vertex z in N (x) which is not dominated by k−1 V (M 0 ) ∪ {w1 , w2 }. Then z ∈ S ∪ (∪i=1 Si ) ∪ Tk . We consider the vertices y, xi , yi as defined above with the supplementary property that if z ∈ Si (respectively, z ∈ Tk ) then yi = z (respectively, yk = z). If z 6∈ S, then {x1 , y1 , . . . , xk , yk , x, y} is a paired-dominating set of G and if z ∈ S, then {x2 , y2 , . . . , xk , yk , y, z, x1 , x} is a paired-dominating set of G. In any case the relations (∗) and (∗∗) may be replaced by γ pr (G) ≤ 2k + 2 and γ pr (G) ≤ δ(G) − 2|W | + 3 = δ(G) − 2|V (M 0 )| + 1. By Claim 6, γ pr (G) + γ pr (G) ≤ δ(G) − |V (M 0 )| + 5 ≤ δ(G) + 3. By symmetry between G and G, γ pr (G) + γ pr (G) ≤ min{δ(G), δ(G)} + 3 and the proof is complete.
Corollary 3. For any graph G of order n ≥ 6 with γ pr (G), γ pr (G) > 4, γ pr (G) + γ pr (G) ≤
n+5 2n − 2 ≤ . 2 3
Proof. Since δ(G) ≤ n −1−δ(G), we have min{δ(G), δ(G)} ≤ δ(G)+δ(G) ≤ n−1 2 2 and 12 ≤ γ pr (G)+γ pr (G) ≤ n+5 n+5 2n−2 by Lemma 2. Moreover 2 ≥ 12 implies n ≥ 19 and thus 2 ≤ 3 .
n+5 2
Theorem 4. Let G be a graph of order n ≥ 6 such that δ(G) ≥ 2 and δ(G) ≥ 2. Then γ pr (G)+γ pr (G) ≤ 2n 3 +4 and 2n+8 the only extremal graphs are B1 and F1 = B1 . If moreover n ≥ 14, then γ pr (G) + γ pr (G) ≤ 3 if n ∈ {14, 17, 20} 2n and γ pr (G) + γ pr (G) ≤ 2n 3 + 2 otherwise. For n ≥ 25, equality γ pr (G) + γ pr (G) = 3 + 2 occurs if and only if each component of G or G belongs to {C6 , C9 , D5,5 } ∪ {Fi |2 ≤ i ≤ 6}.
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Proof. If γ pr (G) = 2 or γ pr (G) = 2, in particular if G or G is not connected, then γ pr (G) + γ pr (G) ≤ 2n 3 + 2 by Theorem D. If γ pr (G) ≥ 6 and γ pr (G) ≥ 6 then γ pr (G) + γ pr (G) ≤ 2n−2 by Corollary 3. Assume now without 3 loss of generality that G and G are connected, γ pr (G) ≥ 4 and γ pr (G) = 4. By Theorem C, G has diameter 2. By 2n Theorem D, γ pr (G) + γ pr (G) ≤ 2n 3 + 4 with equality if and only if γ pr (G) = 3 and γ pr (G) = 4, i.e., G ∈ {B1 , F1 } by Corollary 1. If moreover n ≥ 14 then by Theorem E(2), the even integer γ pr (G) is less than 2n−2 and thus at 3 n−2 n most equal to 2b 3 c. On the other hand, γ pr (G) ≤ 2γ (G) ≤ 2b 4 c + 2 by Theorem B. Note that for n ≥ 14, n n−2 2n+8 2b n4 c + 2 ≤ 2b n−2 3 c except for n = 16. When n ≥ 14, one can check that min{2b 4 c + 2, 2b 3 c} + 4 ≤ 3 , and 2n+6 that min{2b n4 c + 2, 2b n−2 3 c} + 4 ≤ 3 if moreover n 6∈ {14, 17, 20}. This completes the proof. 2n If γ pr (G) + γ pr (G) = 3 + 2 then from the previous proof and without loss of generality, either γ pr (G) = 2 and γ pr (G) = 2n 3 , i.e., each component of G belongs to {C 6 , C 9 , D5,5 } ∪ {Fi |2 ≤ i ≤ 6} by Corollary 1; or γ pr (G) = 4 n 2n−6 and γ pr (G) = 2n 3 − 2 which is impossible for n ≥ 25 by Theorem B since n ≥ 25 implies 2b 4 c + 2 < 3 . Acknowledgement The third author’s research was supported by the Research Office of the Azarbaijan University of Tarbiat Moallem. References [1] J.E. Dunbar, T.W. Haynes, S.T. Hedetniemi, Nordhaus–Gaddum bounds for domination in graphs with specified minimum degree, Util. Math. 67 (2005) 97–105. [2] T.W. Haynes, S.T. Hedetniemi, P.J. Slater, Fundamentals of Domination in Graphs, Marcel Decker, Inc., 1998. [3] T.W. Haynes, P.J. Slater, Paired domination in graphs, Networks 32 (1998) 199–206. [4] T.W. Haynes, P.J. Slater, Paired domination and paired-domatic number, Congr. Numer. 109 (1995) 65–72. [5] M.A. Henning, Graphs with large paired-domination number, J. Combin. Optim. 13 (2007) 61–78. [6] A. Hellwig, L. Volkmann, Some upper bounds for the domination number, J. Combin. Math. Combin. Comput. 57 (2006) 187–209. [7] J.P. Joseph, S. Arumugam, Domination in graphs, Internat. J. Management Systems 11 (1995) 177–182. [8] F. Jaeger, C. Payan, Relations du type Nordhaus–Gaddum pour le nombre d’absorption d’un graphe simple, C. R. Acad. Sci. Paris S´er. A 274 (1972) 728–730. [9] D.B. West, Introduction to Graph Theory, Prentice-Hall, Inc., 2000.
Note
Paired-domination number of a graph and its complement O. Favaron a,b , H. Karami c , S.M. Sheikholeslami d,∗ a Univ Paris-Sud, LRI, UMR 8623, Orsay, F-91405, France b CNRS, Orsay, F-91405, France c Department of Mathematics, Sharif University of Technology, P.O. Box 11365-9415, Tehran, Islamic Republic of Iran d Department of Mathematics, Azarbaijan University of Tarbiat Moallem, Tabriz, Islamic Republic of Iran
Received 18 July 2007; received in revised form 15 November 2007; accepted 18 November 2007 Available online 26 December 2007
Abstract A paired-dominating set of a graph G = (V, E) with no isolated vertex is a dominating set of vertices inducing a graph with a perfect matching. The paired-domination number of G, denoted by γ pr (G), is the minimum cardinality of a paired-dominating set of G. We consider graphs of order n ≥ 6, minimum degree δ such that G and G do not have an isolated vertex and we prove that – if γ pr (G) > 4 and γ pr (G) > 4, then γ pr (G) + γ pr (G) ≤ 3 + min{δ(G), δ(G)}. 2n – if δ(G) ≥ 2 and δ(G) ≥ 2, then γ pr (G) + γ pr (G) ≤ 2n 3 + 4 and γ pr (G) + γ pr (G) ≤ 3 + 2 if moreover n ≥ 21. c 2007 Elsevier B.V. All rights reserved.
Keywords: Paired-domination number; Nordhaus–Gaddum inequalities
1. Introduction We use [2,9] for the terminology and notation which are not defined here. Let G = (V (G), E(G)) be a graph of order n with no isolated vertex. The (closed) neighborhood of a vertex u is denoted by N G (u) (N G [u]) and its degree |N G (u)| by dG (u) (briefly N (u) (N [u]) and d(u) when no ambiguity on the graph is possible). A dominating set of a graph G is a set S of vertices such that every vertex of V \ S is adjacent to some vertex in S. The domination number of G, denoted by γ (G), is the minimum cardinality of a dominating set. A paired-dominating set (PDS) of G is a dominating set S such that the subgraph G[S] induced by S contains a perfect matching (not necessarily induced). Every graph without isolated vertices has a paired-dominating set since the end-vertices of any maximal matching form such a set. The paired-domination number of G, denoted by γ pr (G), is the minimum cardinality of a PDS. When G is not connected, let G 1 , . . . , G k be its components. Then γ pr (G) exists if each G i has order at least 2, and Pk γ pr (G) = i=1 γ pr (G i ). A paired-dominating set of cardinality γ pr (G) is called a γ pr (G)-set. Paired-domination number was introduced by Haynes and Slater [4,3] as a model for assigning backups to guards for security purposes. They observed in [3] that γ pr (G) is even and γ pr (G) ≤ 2γ (G) for every graph with δ(G) ≥ 1. The complement G of a graph G has vertex set V (G) and x y ∈ E(G) if and only if x y 6∈ E(G). For any graph parameter µ, bounds on ∗ Corresponding author.
E-mail addresses: [email protected] (O. Favaron), [email protected] (S.M. Sheikholeslami). c 2007 Elsevier B.V. All rights reserved. 0012-365X/$ - see front matter doi:10.1016/j.disc.2007.11.034
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Fig. 1. Graphs whose paired-domination is 2n 3 .
µ(G) + µ(G) and on µ(G)µ(G) are called Nordhaus–Gaddum inequalities. Many Nordhaus–Gaddum bounds have been obtained on various domination parameters. Some of them can be improved when additional constraints on G and G are added. For instance it has been proved Theorem A. 1. γ (G) + γ (G) ≤ n + 1 for every graph G [8]; 2. γ (G) + γ (G) ≤ n2 + 2 if δ(G), δ(G) ≥ 1 [7]; 3. γ (G) + γ (G) ≤ 2n 5 + 3 if δ(G), δ(G) ≥ 2 with some small exceptions [1]; 3n 4. γ (G) + γ (G) ≤ 8 + 2 if δ(G), δ(G) ≥ 3 with some small exceptions [1]. In what concerns paired-domination, we must suppose δ(G), δ(G) ≥ 1. From Theorem A(2) and the fact that γ pr (G) ≤ 2γ (G), Haynes and Slater immediately obtained the bound γ pr (G) + γ pr (G) ≤ 2(b n2 c + 2) for every graph G with δ(G), δ(G) ≥ 1 [3]. The same argument applied to Theorem A(3) shows that, except for some small graphs, γ pr (G) + γ pr (G) ≤ 2(b 2n 5 c + 3) for every graph G with δ(G), δ(G) ≥ 2. Our purpose in this paper is to improve this bound. We will use the following known results on γ (G) and γ pr (G). Theorem B (Hellwig and Volkmann [6]). Let G be graph of order n and diameter 2. Then γ (G) ≤ b n4 c + 1. Theorem C (Haynes and Slater [3]). For any graph G, γ pr (G) ≥ 4 if and only if diam(G) = 2. Theorem D (Haynes and Slater [3]). If G is a connected graph of order n ≥ 6 with δ(G) ≥ 2, then γ pr (G) ≤
2n 3 .
Henning improved Theorem D by determining the extremal or nearly extremal graphs. Theorem E (Henning [5]). Let G be a connected graph of order n ≥ 6 with δ(G) ≥ 2. 2n−2 1. γ pr (G) = 2n for 3 if and only if G is C 6 , C 9 or one of the graphs illustrated in Fig. 1. Hence γ pr (G) ≤ 3 n ≥ 10. 2. Moreover if n ≥ 14 then γ pr (G) = 2n−2 3 if and only if G belongs to a well-determined family H of graphs of diameter greater than 2.
We note that F1 = B1 and γ pr (B1 ) = γ pr (F1 ) = 4, and that if G ∈ {C6 , C9 , D5,5 } ∪ {Fi |2 ≤ i ≤ 6} then diam(G) > 2. An immediate consequence follows from Theorem E(1) and Theorem C. Corollary 1. Let G be a connected graph of order n ≥ 6 with δ(G), δ(G) ≥ 2 and γ pr (G) = 2n then γ pr (G) + γ pr (G) = 2n 3 + 2 and if G ∈ {B1 , F1 } then γ pr (G) + γ pr (G) = 3 + 4.
2n 3 .
If G 6∈ {B1 , F1 }
2. A bound on the sum γ pr (G) + γ pr (G) 2n 3
In this section we prove that if G is a simple graph of order n ≥ 6 with δ(G), δ(G) ≥ 2, then γ pr (G) + γ pr (G) ≤ + 4. If moreover n ≥ 21, then γ pr (G) + γ pr (G) ≤ 2n 3 + 2.
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Lemma 2. For any graph G of order n ≥ 6 with γ pr (G), γ pr (G) > 4, γ pr (G) + γ pr (G) ≤ min{δ(G), δ(G)} + 3. Proof. Since the paired-domination number is always even, γ pr (G), γ pr (G) ≥ 6. We consider a vertex x of degree δ in G and X = V \ N [x]. If N (x) ∩ N (y) = ∅ for some y ∈ X then in G, x dominates X , y dominates N (x), and {x, y} is a paired-dominating set of G, a contradiction. Therefore, N (x) ∩ N (y) 6= ∅ for all y ∈ X. Hence, N (x) dominates X but no vertex of N (x) dominates X for otherwise γ pr (G) = 2. The first claim is an immediate consequence of γ pr (G) ≥ 6. Claim 1. Let A be a subset of N (x) dominating X and B a maximal subset of A not dominating X . Then |A| ≥ 3 and |B| ≥ 2. Let S be a maximum subset of N (x) which does not dominate all vertices in X and T = N (x) \ S. By the maximality of S, every vertex t of T dominates X \ N (S) and by Claim 1, |S| ≥ 2. Claim 2. T dominates X . Proof of Claim 2. Let, to the contrary, y ∈ X be not dominated by T . By the definition of S, there exists a vertex z ∈ X not dominated by S. Since y ∈ N (S) and z ∈ X − N (S), y 6= z. Now {x, z, y, w} is a paired-dominating set of G for each w ∈ T , a contradiction. Claim 3. T has no isolated vertex. Proof of Claim 3. Let, to the contrary, y be an isolated vertex in T . By Claim 1, y does not dominate X . Let z ∈ X be not dominated by y and w ∈ X \ N (S). Obviously z ∈ N (S) and thus w 6= z. Now {x, w, y, z} is a paired-dominating set of G, a contradiction. Let M = {x1 y1 , x2 y2 , . . . , xs ys } be a maximal matching of G[T ]. Then s ≥ 1 by Claim 3 and U = T \ V (M) is an independent set of G. Claim 4. If V (M) does not dominate X , then V (M) ∪ {y} dominates X for all y ∈ T \ V (M). Proof of Claim 4. If V (M) does not dominate X , then U = T \ V (M) 6= ∅ by Claim 2. Suppose V (M) ∪ {y} does not dominate X for some y ∈ U and let w1 ∈ X be not dominated by V (M) ∪ {y}. Let w2 ∈ X \ N (S). In G, w1 dominates V (M), y dominates U since U is independent in G, and {x, w2 , y, w1 } is a paired-dominating set of G, a contradiction. Let M 0 be a maximal submatching of M such that V (M 0 ) does not dominate X and let W be a maximum subset of T containing V (M 0 ) and not dominating X . By Claims 1 and 2 and the definition of S, M 0 6= ∅, W 6= T and |S| ≥ |W |. However it is possible that M 0 = M, in which case W = V (M) by Claim 4, or that V (M 0 ) ⊆ W 6= V (M). Claim 5. If V (M 0 ) ∪ {y} dominates X for some y ∈ N (x) \ V (M 0 ), in particular if M 0 = M or if W = V (M 0 ), then γ pr (G) ≤ |V (M 0 )| + 2 and |V (M 0 )| ≥ 4. Proof of Claim 5. If V (M 0 ) ∪ {y} dominates X , then V (M 0 ) ∪ {x, y} is a paired-dominating set of G, which proves the claim since γ pr (G) ≥ 6. Claim 6. In any case, γ pr (G) ≤ |V (M 0 )| + 4. Proof of Claim 6. If M 0 = M then γ pr (G) ≤ |V (M 0 )| + 2 by Claim 5. Otherwise, let xs ys ∈ M − M 0 . By the maximality of M 0 , V (M 0 ) ∪ {xs , ys } dominates X and V (M 0 ) ∪ {xs , ys , x, a}, where a is any vertex of S, is a paireddominating set of G.
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Let T1 = T \ W . If T1 dominates X , let S1 be a maximal subset of T1 not dominating X and T2 = T1 \ S1 . If T2 dominates X , let S2 be a maximal subset of T2 not dominating X and T3 = T2 \ S2 . We continue the process until obtention of a subset Tk of T not dominating X . Thus we construct a finite chain T = T0 % T1 % · · · % Tk (k ≥ 1) such that (a) For 0 ≤ i ≤ k − 1, Ti dominates X . (b) For 0 ≤ i ≤ k − 1, Si = Ti \ Ti+1 does not dominate X , but Si ∪ {y} dominates X for each y ∈ Ti+1 . (c) Tk does not dominate X . Then S0 = W and if k ≥ 2, |Si | ≥ 2 for 1 ≤ i ≤ k − 1 by Claim 1. For all i between 0 and k − 1, let xi be a vertex of X not dominated by Si and let yi ∈ Si . Let xk be a vertex of X not dominated by Tk and yk ∈ Tk . By (b) the vertex xi is adjacent to every vertex in Ti+1 . Hence all the vertices xi are distinct. Similarly, all the vertices yi are distinct since S0 , S1 ,..., Sk−1 , Tk are disjoint. Finally let y ∈ X \ N (S). Then {x0 , y0 , x1 , y1 , . . . , xk , yk , x, y} is a paired-dominating set of G and thus γ pr (G) ≤ 2k + 4.
(∗)
On the other hand, δ(G) = |S|+|W |+ and by (∗),
Pk−1 i=1
|Ti \Ti+1 |+|Tk | ≥ 2|W |+2(k −1)+1. Therefore 2k ≤ δ(G)−2|W |+1
γ pr (G) ≤ δ(G) − 2|W | + 5.
(∗∗)
If |W | ≥ |V (M 0 )| + 2 then γ pr (G) ≤ δ(G) − 2|V (M 0 )| + 1 and by Claim 6, γ pr (G) + γ pr (G) ≤ δ(G) − |V (M 0 )| + 5 ≤ δ(G) + 3. If |W | = |V (M 0 )| + 1 and V (M 0 ) ∪ {y} dominates X for some y ∈ T \ V (M 0 ), or if W = V (M 0 ), then by Claim 5, γ pr (G) ≤ |V (M 0 )| + 2 ≤ |W | + 2 and |W | ≥ 4. Therefore by (∗∗), γ pr (G) + γ pr (G) ≤ δ(G) − |W | + 7 ≤ δ(G) + 3. If |W | = |V (M 0 )| + 1 and V (M 0 ) ∪ {y} does not dominate X for any y ∈ T \ V (M 0 ), let w1 w2 ∈ M \ M 0 . By the definition of M 0 , V (M 0 ) ∪ {w1 , w2 } dominates X and without loss of generality we may assume that W = V (M 0 ) ∪ {w1 }. If V (M 0 ) ∪ {w1 , w2 } dominates N (x), then γ pr (G) ≤ |V (M 0 )| + 2 and, as before, we have γ pr (G) + γ pr (G) ≤ δ(G) + 3. Finally suppose there exists a vertex z in N (x) which is not dominated by k−1 V (M 0 ) ∪ {w1 , w2 }. Then z ∈ S ∪ (∪i=1 Si ) ∪ Tk . We consider the vertices y, xi , yi as defined above with the supplementary property that if z ∈ Si (respectively, z ∈ Tk ) then yi = z (respectively, yk = z). If z 6∈ S, then {x1 , y1 , . . . , xk , yk , x, y} is a paired-dominating set of G and if z ∈ S, then {x2 , y2 , . . . , xk , yk , y, z, x1 , x} is a paired-dominating set of G. In any case the relations (∗) and (∗∗) may be replaced by γ pr (G) ≤ 2k + 2 and γ pr (G) ≤ δ(G) − 2|W | + 3 = δ(G) − 2|V (M 0 )| + 1. By Claim 6, γ pr (G) + γ pr (G) ≤ δ(G) − |V (M 0 )| + 5 ≤ δ(G) + 3. By symmetry between G and G, γ pr (G) + γ pr (G) ≤ min{δ(G), δ(G)} + 3 and the proof is complete.
Corollary 3. For any graph G of order n ≥ 6 with γ pr (G), γ pr (G) > 4, γ pr (G) + γ pr (G) ≤
n+5 2n − 2 ≤ . 2 3
Proof. Since δ(G) ≤ n −1−δ(G), we have min{δ(G), δ(G)} ≤ δ(G)+δ(G) ≤ n−1 2 2 and 12 ≤ γ pr (G)+γ pr (G) ≤ n+5 n+5 2n−2 by Lemma 2. Moreover 2 ≥ 12 implies n ≥ 19 and thus 2 ≤ 3 .
n+5 2
Theorem 4. Let G be a graph of order n ≥ 6 such that δ(G) ≥ 2 and δ(G) ≥ 2. Then γ pr (G)+γ pr (G) ≤ 2n 3 +4 and 2n+8 the only extremal graphs are B1 and F1 = B1 . If moreover n ≥ 14, then γ pr (G) + γ pr (G) ≤ 3 if n ∈ {14, 17, 20} 2n and γ pr (G) + γ pr (G) ≤ 2n 3 + 2 otherwise. For n ≥ 25, equality γ pr (G) + γ pr (G) = 3 + 2 occurs if and only if each component of G or G belongs to {C6 , C9 , D5,5 } ∪ {Fi |2 ≤ i ≤ 6}.
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Proof. If γ pr (G) = 2 or γ pr (G) = 2, in particular if G or G is not connected, then γ pr (G) + γ pr (G) ≤ 2n 3 + 2 by Theorem D. If γ pr (G) ≥ 6 and γ pr (G) ≥ 6 then γ pr (G) + γ pr (G) ≤ 2n−2 by Corollary 3. Assume now without 3 loss of generality that G and G are connected, γ pr (G) ≥ 4 and γ pr (G) = 4. By Theorem C, G has diameter 2. By 2n Theorem D, γ pr (G) + γ pr (G) ≤ 2n 3 + 4 with equality if and only if γ pr (G) = 3 and γ pr (G) = 4, i.e., G ∈ {B1 , F1 } by Corollary 1. If moreover n ≥ 14 then by Theorem E(2), the even integer γ pr (G) is less than 2n−2 and thus at 3 n−2 n most equal to 2b 3 c. On the other hand, γ pr (G) ≤ 2γ (G) ≤ 2b 4 c + 2 by Theorem B. Note that for n ≥ 14, n n−2 2n+8 2b n4 c + 2 ≤ 2b n−2 3 c except for n = 16. When n ≥ 14, one can check that min{2b 4 c + 2, 2b 3 c} + 4 ≤ 3 , and 2n+6 that min{2b n4 c + 2, 2b n−2 3 c} + 4 ≤ 3 if moreover n 6∈ {14, 17, 20}. This completes the proof. 2n If γ pr (G) + γ pr (G) = 3 + 2 then from the previous proof and without loss of generality, either γ pr (G) = 2 and γ pr (G) = 2n 3 , i.e., each component of G belongs to {C 6 , C 9 , D5,5 } ∪ {Fi |2 ≤ i ≤ 6} by Corollary 1; or γ pr (G) = 4 n 2n−6 and γ pr (G) = 2n 3 − 2 which is impossible for n ≥ 25 by Theorem B since n ≥ 25 implies 2b 4 c + 2 < 3 . Acknowledgement The third author’s research was supported by the Research Office of the Azarbaijan University of Tarbiat Moallem. References [1] J.E. Dunbar, T.W. Haynes, S.T. Hedetniemi, Nordhaus–Gaddum bounds for domination in graphs with specified minimum degree, Util. Math. 67 (2005) 97–105. [2] T.W. Haynes, S.T. Hedetniemi, P.J. Slater, Fundamentals of Domination in Graphs, Marcel Decker, Inc., 1998. [3] T.W. Haynes, P.J. Slater, Paired domination in graphs, Networks 32 (1998) 199–206. [4] T.W. Haynes, P.J. Slater, Paired domination and paired-domatic number, Congr. Numer. 109 (1995) 65–72. [5] M.A. Henning, Graphs with large paired-domination number, J. Combin. Optim. 13 (2007) 61–78. [6] A. Hellwig, L. Volkmann, Some upper bounds for the domination number, J. Combin. Math. Combin. Comput. 57 (2006) 187–209. [7] J.P. Joseph, S. Arumugam, Domination in graphs, Internat. J. Management Systems 11 (1995) 177–182. [8] F. Jaeger, C. Payan, Relations du type Nordhaus–Gaddum pour le nombre d’absorption d’un graphe simple, C. R. Acad. Sci. Paris S´er. A 274 (1972) 728–730. [9] D.B. West, Introduction to Graph Theory, Prentice-Hall, Inc., 2000.
