We investigate several versions of restraineddomination numbers and present new bounds on these parameters. We generalize theconcept of restrained domination and improve some well-known bounds in the literature.In particular, for a graph $G$ of order $n$ and minimum degree $\delta\geq 3$, we prove thatthe restrained double domination number of $G$ is at most $n-\delta+1$. In addition,for a connected cubic graph $G$ of order $n$ we show thatthe total restrained domination number of $G$ is at least $n/3$ andthe restrained double domination number of $G$ is at least $n/2$

Golmohammadi, Hamid R

Khodkar, Abdollah

Samadi, Babak

English

Contributions to Discrete Mathematics (E-Journal, University of Calgary)

Volume 12, Number 1, Pages 14–19ISSN 1715-0868BOUNDS ON SEVERAL VERSIONS OF RESTRAINEDDOMINATION NUMBERSBABAK SAMADI, HAMID R. GOLMOHAMMADI, AND ABDOLLAH KHODKARAbstract. We investigate several versions of restrained dominationnumbers and present new bounds on these parameters. We general-ize the concept of restrained domination and improve some well-knownbounds in the literature. In particular, for a graph G of order n andminimum degree δ ≥ 3, we prove that the restrained double dominationnumber of G is at most n − δ + 1. In addition, for a connected cubicgraph G of order n, we show that the total restrained domination num-ber of G is at least n/3 and the restrained double domination numberof G is at least n/2.1. IntroductionThroughout this paper, let G be a finite connected graph with vertex setV = V (G), edge set E = E(G), minimum degree δ = δ(G), and maximumdegree ∆ = ∆(G). We use [13] for terminology and notation which are notdefined here. For any vertex v ∈ V , N(v) = {u ∈ G | uv ∈ E(G)} denotesthe open neighbourhood of v in G, and N [v] = N(v)∪ {v} denotes its closedneighbourhood. A set S ⊆ V (G) is a dominating set in G if each vertex inV \ S is adjacent to at least one vertex in S. A dominating set S is a re-strained dominating set (total restrained dominating set) if the subgraph(s)induced by the vertices of V \S (V \S and S) has (have) no isolated vertices.The restrained domination number γr(G) (total restrained domination num-ber γrt (G)) is the minimum cardinality of a restrained dominating set (totalrestrained dominating set). The concept of restrained domination was intro-duced by Telle and Proskurowski [12], while its total version was introducedby Ma, Chen and Sun [11].A generalization of the concept of total restrained domination was ex-hibited in [10] as k-tuple total restrained domination. The k-tuple totalrestrained domination number, γr×k,t(G), of a graph G is the minimum car-dinality of a k-tuple total restrained dominating set as a subset S ⊆ V withthis property that every vertex in V has at least k neighbors in S and everyReceived by the editors November 30, 2014, and in revised form April 12, 2015.2000 Mathematics Subject Classification. 05C69.Key words and phrases. restrained domination, restrained double domination, totalrestrained domination.c©2017 University of Calgary14BOUNDS ON SEVERAL VERSIONS OF RESTRAINED DOMINATION NUMBERS 15vertex in V \S has at least k neighbors in V \S. This concept was introducedby Kazemi in [10].Kala and Nirmala Vasantha [9] defined the concept of restrained doubledomination. In fact, a set S ⊆ V (G) is a restrained double dominating setin G if every vertex in V is dominated by at least two vertices in S andthe subgraph induced by V \ S has no isolated vertices. The minimumcardinality of a restrained double dominating set is called the restraineddouble domination number of G and is denoted by γ2r(G).In this paper, we first obtain some bounds on γrt (G) and γ2r(G) of agraph G that improve their corresponding results in [1, 2, 8, 9, 11]. Then fora graph G we introduce the concept of the (k, k′, k′′)-domination number,denoted γ(k,k′,k′′)(G), as a generalization of the parameters γr(G), γrt (G),γ2r(G) and γr×k,t(G). Finally, we find a lower bound for γ(k,k′,k′′)(G). Thisbound improves some of the known results presented in [2, 3, 4, 6, 7, 10]. Inparticular, for a cubic graph G we conclude γr(G) ≥ n/4 (see [5]), γrt (G) ≥n/3 and γ2r(G) ≥ n/2.2. Bounds on γrt (G) and γ2r(G)We make use of the following lemma to show that the bounds found inthis section are sharp.Lemma 2.1. Let m and n be positive integers. Then(i) γrt (Km,n) ={m+ n, if min{m,n} = 1,2, otherwise,(ii) γrt (Pn) = n− 2bn−24 c,(iii) γ2r(Kn) ={2, if n 6= 3,3, if n = 3,(iv) γ2r(K1,n) = n+ 1.See [2] for (i), (ii) and see [9] for (iii), (iv).The following theorem gives lower bounds on the total restrained andrestrained double domination numbers of a graph G with ` leaves and ssupport vertices.Theorem 2.2. Let G be a graph of order n without isolated vertices. Sup-posed that G has ` leaves and s support vertices. Then(i)γrt (G) ≥ `+ max{n− s∆,n∆ + 1},(ii)γ2r(G) ≥ max{2n+ (∆ + 1)`∆ + 2,2n+ `∆ + (∆− 2)n2 − s∆ + 1},where n2 is the number of vertices of degree two that are not supportvertices.16 BABAK SAMADI, HAMID R. GOLMOHAMMADI, AND ABDOLLAH KHODKARIn addition, these bounds are sharp.Proof. To prove (i), we first prove the following claims.Claim 1. γrt (G) ≥ `+ (n− s)/∆.Let S be a total restrained dominating set in G. Let L be the set of leavesand let {wi}si=1 be the set of support vertices. Suppose li is the number ofleaves adjacent to wi for every 1 ≤ i ≤ s. Since all leaves and supportvertices belong to S, it follows that no leaf has neighbors in V \ S and eachwi has at most ∆ − `i neighbors in V \ S, for every 1 ≤ i ≤ s. Also everyvertex in S \ (L ∪ {wi}si=1) has at most ∆− 1 neighbors in V \ S. Now weconsider the set [S, V \S] of edges with a vertex in S and a vertex in V \S.We have(2.1)s∑i=1(∆− `i) + (|S| − `− s)(∆− 1) ≥ |[S, V \ S]|.On the other hand, every vertex in V \ S has at least one neighbor in S.Hence(2.2) |[S, V \ S]| ≥ n− |S|.Since `1 + ...+ `s = `, the inequalities (2.1) and (2.2) implys∆− `+ (|S| − `− s)(∆− 1) ≥ n− |S|.This implies the desired lower bound.Claim 2. γrt (G) ≥ `+ n/(∆ + 1).Let S and L be as in the proof of Claim 1. It is easy to check that S \ Lis a dominating set in G. Suppose α is the number of edges with at leastone vertex in S \ L. Then(n− |S|) + ` ≤ α ≤ ∆(|S| − `).This completes the proof of Claim 2.The result now follows from Claim 1 and Claim 2. The bound is sharpfor P 4n+2 and the star K1,n by Lemma 2.1. This completes the proof of (i).The proof of (ii) is similar to the proof of (i). Note that all leaves, sup-port vertices and vertices of degree two belong to every restrained doubledominating set. The bound is sharp for the star K1,n. In [1, 8, 11], it was proved that γrt (T ) ≥ ∆(T )+1, where T is a tree. Notethat ` + (n − s)/∆ ≥ ∆(T ) + 1 and ` + n/(∆ + 1) ≥ ∆(T ) + 1. Thus thelower bound in (i) of Theorem 2.2 is tighter than its corresponding boundgiven in [1, 8, 11].It is easy to check that the lower bound given in (ii) of Theorem 2.2 isan improvement of the lower bound γ2r(G) ≥ 2n/(∆ + 1), which was firstgiven in [9] for a graph G of order n without isolated vertices.BOUNDS ON SEVERAL VERSIONS OF RESTRAINED DOMINATION NUMBERS 17We conclude this section by establishing an upper bound on the restraineddouble domination number of a graph in terms of its order and its minimumdegree.Theorem 2.3. If G is a graph of order n and minimum degree δ ≥ 3, thenγ2r(G) ≤ n− δ + 1 and the bound is sharp.Proof. Let u be a vertex in G with deg(u) = δ and v, w ∈ N(u). Sinceδ ≥ 3, it follows that |N [u] \ {v, w}| ≥ 2 and N [u] \ {v, w} is a nonemptyset. Also, it is easy to see that the subgraph induced by N [u] \ {v, w} hasno isolated vertices. Now let S = V (G) \ (N [u] \ {v, w}). Obviously, u isadjacent to both v and w, and v, w ∈ S. Let x ∈ N(u) \ {v, w}. Then x canbe joined to at most δ− 2 vertices in N [u] \ {v, w}. Thus x has at least twoneighbors in S. Finally, no vertex in S has all its neighbors in N [u] \ {v, w}because |N [u] \ {v, w}| = δ − 1. Thus, there are no isolated vertices in thesubgraph induced by S. Therefore S is a restrained double dominating setin G. Hence,γ2r(G) ≤ |S| = |V (G)− (N [u] \ {v, w})| = n− δ + 1.The upper bound is sharp for the complete graph Kn when n ≥ 4, by Lemma2.1. The authors in [9] showed that γ2r(G) ≤ n− 2 for every graph of order nand minimum degree δ(G) ≥ 3. In fact, Theorem 2.3 gives an improvementfor this bound. In addition, if δ ≥ 4, then the upper bound n− 2 for γ2r(G)is not sharp by Theorem 2.3.3. (k, k′, k′′)-dominating setsLet k, k′ and k′′ be nonnegative integers. A set S ⊆ V is a (k, k′, k′′)-dominating set in G if every vertex in S has at least k neighbors in S, everyvertex in V \ S has at least k′ neighbors in S, and at least k′′ neighborsin V \ S. The (k, k′, k′′)-domination number γ(k,k′,k′′)(G) is the minimumcardinality of a (k, k′, k′′)-dominating set. We note that every graph with theminimum degree at least k has a (k, k′, k′′)-dominating set, since S = V (G)is such a set. Note thatγ(0,1,1)(G) = γr(G), γ(1,1,1)(G) = γrt (G),γ(1,2,1)(G) = γ2r(G), γ(k,k,k)(G) = γr×k,t(G).In this section, we calculate a lower bound on γ(k,k′,k”)(G), which improvesthe existing lower bounds on these four parameters.Theorem 3.1 ([2, 4]). If G is a graph without isolated vertices of order nand size m, then(3.1) γrt (G) ≥ 3n/2−m.In addition this bound is sharp.18 BABAK SAMADI, HAMID R. GOLMOHAMMADI, AND ABDOLLAH KHODKARAlso Hattingh et al. [6] found that(3.2) γr(G) ≥ n− 2m/3.The following known result is an immediate consequence of Theorem 3.1.Theorem 3.2 ([3]). If T is a tree of order n ≥ 2, then(3.3) γrt (T ) ≥⌈n+ 22⌉.The inequality(3.4) γr(T ) ≥⌈n+ 23⌉on restrained domination number of a tree of order n ≥ 1 was obtained byHattingh and Rautenbach [7].The author in [10] generalized Theorem 3.1 and proved that if δ(G) ≥ k,then(3.5) γr×k,t(G) ≥ 3n/2−m/k.Moreover the authors in [9] proved that if G is a graph without isolatedvertices, then(3.6) γ2r(G) ≥ 5n− 2m4.We now improve the lower bounds given in inequalities (3.1)–(3.6). Forthis purpose we introduce a notation. Let G be a graph with δ(G) ≥ k andS be a (k, k′, k′′)-dominating set in G. We defineδ∗ = min{deg(v) | v ∈ V (G) and deg(v) ≥ k′ + k′′}.It is easy to see that if v ∈ V \ S, then deg(v) is at least k′ + k′′ andtherefore is at least δ∗.Theorem 3.3. Let G be a graph with δ(G) ≥ k. Thenγ(k,k′,k′′)(G) ≥(k′ + δ∗)n− 2mδ∗ + k′ − k .Proof. Let S be a minimum (k, k′, k′′)-dominating set in G. Then, everyvertex v ∈ S is adjacent to at least k vertices in S. Therefore |E(G[S])| ≥k|S|/2. Let E(v) be the set of edges at vertex v. Now let v ∈ V \S. Since Sis a (k, k′, k′′)-dominating set, it follows that v is incident to at least k′ edgese1, ..., ek′ in [S, V \ S] and at least k′′ edges ek′+1, ..., ek′+k′′ in E(G[V \ S]).Since deg(v) ≥ δ∗ ≥ k′ + k′′, v is incident to at least δ∗ − k′ − k′′ edges inE(v) \ {ei}k′+k′′i=1 . The value of |[S, V \S]|+ |E(G[V \S])| is minimized if theedges in E(v) \ {ei}k′+k′′i=1 belong to E(G[V \ S]). Therefore2m = 2|E(G[S])|+ 2|[S, V \ S]|+ 2|E(G[V \ S])|≥ k|S|+ 2k′(n− |S|) + k′′(n− |S|) + (δ∗ − k′ − k′′)(n− |S|).This leads to γ(k,k′,k′′)(G) = |S| ≥ ((k′ + δ∗)n− 2m)/(δ∗ + k′ − k). BOUNDS ON SEVERAL VERSIONS OF RESTRAINED DOMINATION NUMBERS 19We note that when (k, k′, k′′) = (1, 1, 1), then Theorem 3.3 gives improve-ments for inequalities (3.1) and (3.3). When (k, k′, k′′) = (0, 1, 1), then itwill be improvements of its corresponding inequalities given by (3.2) and(3.4). Also, if (k, k′, k′′) = (k, k, k), Theorem 3.3 improves inequality (3.5)and if (k, k′, k′′) = (1, 2, 1), it improves inequality (3.6).The following result of Hattingh and Joubert is an immediate consequenceof Theorem 3.3.Corollary 3.4 ([5]). If G is a cubic graph of order n, then γr(G) ≥ n/4.Also, for total restrained and restrained double domination numbers of acubic graph G, we obtain γrt (G) ≥ n/3 and γ2r(G) ≥ n/2 by Theorem 3.3.References1. X. Chen, J. Liu, and J. Meng, Total restrained domination in graphs, Comput. Math.Appl. 62 (2011), 2892–2898.2. J. Cyman and J. Raczek, On the total restrained domination number of a graph,Australas. J. Combin. 36 (2006), 91–100.3. J.H. Hattingh, E. Jonck, E.J. Joubert, and A.R. Plummer, Total restrained dominationin trees, Discrete Math. 307 (2007), 1643–1650.4. , Total restrained domination in unicyclic graphs, Util. Math. 82 (2010), 81–96.5. J.H. Hattingh and E.J. Joubert, Restrained domination in cubic graphs, J. Combin.Optim. 22 (2011), 166–179.6. J.H. Hattingh, E.J. Joubert, M. Loizeaux, A.R. Plummer, and L. Van Der Merwe,Restrained domination in unicyclic graphs, Discuss. Math. 29 (2009), 71–86.7. J.H. Hattingh and D. Rautenbach, Further results on weak domination in graphs, Util.Math. 61 (2002), 193–207.8. N. Jafari Rad, Results on total restrained domination in graphs, Int. J. Contemp.Math. Sci. 3 (2008), 383–387.9. R. Kala and T.R. Nirmala Vasantha, Restrained double domination number of a graph,AKCE Int. J. Graphs Combin. 5 (2008), 73–82.10. A.P. Kazemi, k-tuple total restrained domination/domatic in graphs, Bull. IranianMath. Soc. 40 (2014), 751–763.11. S. Ma, X. Chen, and L. Sun, On total restrained domination in graphs, CzechoslovakMath. J. 55 (2005), 165–173.12. J.A. Telle and A. Proskurowski, Algorithms for vertex partitioning problems on partialk-trees, SIAM J. Discrete Math. 10 (1997), 529–550.13. D.B. West, Introduction to graph theory, 2nd ed., Prentice Hall, USA, 2001.Department of MathematicsUniversity of Mazandaran, Babolsar, IRIE-mail address: samadibabak62@gmail.comDepartment of MathematicsUniversity of Tafresh, Tafresh, IRIE-mail address: h.golmohamadi@tafreshu.ac.irDepartment of MathematicsUniversity of West Georgia Carrollton, GA 30118, USAE-mail address: akhodkar@westga.edu

Bounds on several versions of restrained domination number

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Bounds on several versions of restrained domination number

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Contributions to Discrete Mathematics (E-Journal, University of Calgary)