Restrained domination in graphs

Domke, Gayla S.; Hattingh, Johannes H.; Hedetniemi, Stephen T.; Laskar, Renu C.; Markus, Lisa R.

doi:10.1016/s0012-365x(99)00016-3

Cited by 139 publications

(69 citation statements)

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“…The corresponding result for the domination number was presented by Jaeger and Payan [4]: if G is a graph of order n 2, then γ(G)+γ(G) n+1. A bound on the sum of restrained domination numbers of a graph and its complement was given by G. S. Domke [1]: If G is a graph of order n 2 such that both G and G are not P 3 , then γ r (G) + γ r (G) n + 2.…”

Section: Nordhaus-gaddum-type Results For Total Restrained Dominationmentioning

confidence: 99%

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On total restrained domination in graphs

Chen

Sun

2005

Czech Math J

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Section: Nordhaus-gaddum-type Results For Total Restrained Dominationmentioning

confidence: 99%

“…Graph theory terminology not presented here can be found in [1]. Let G = (V, E) be a graph with |V | = n. The degree, neighborhood and closed neighborhood of a vertex v in the graph G are denoted by d (v), N (v) and N [v] = N (v) ∪ {v}, respectively.…”

Section: Introductionmentioning

confidence: 99%

On total restrained domination in graphs

Chen

Sun

2005

Czech Math J

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“…The restrained domination number γ r (G) is the minimum cardinality of a restrained dominating set of G. The restrained domination number was introduced by Domke et al [14] and has been studied by several author (see for example [12,13]). The restrained bondage number b r (G) of a nonempty graph G is the minimum cardinality among all sets of edges F ⊆ E (G) for which γ r (G −F ) > γ r (G).…”

Section: Introductionmentioning

confidence: 99%

The restrained rainbow bondage number of a graph

Amjadi¹,

Khoeilar²,

Dehgardi³

et al. 2018

Tamkang J. Math.

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Abstract.A restrained k-rainbow dominating function (RkRDF) of a graph G is a function f from the vertex set V (G) to the set of all subsets of the set {1, 2, . . . , k} such that forThe minimum weight of a restrained k-rainbow dominating function of G is called the restrained k-rainbow domination number of G, denoted by γ r r k (G). The restrained krainbow bondage number b r r k (G) of a graph G with maximum degree at least two is the minimum cardinality of all sets F ⊆ E (G) for which γ r r k (G − F ) > γ r r k (G). In this paper, we initiate the study of the restrained k-rainbow bondage number in graphs and we present some sharp bounds for b r r 2 (G). In addition, we determine the restrained 2-rainbow bondage number of some classes of graphs.

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“…In this article, we study a variation on the domination theme, which is called restrained domination, introduced by Telle and Proskurowski [14], albeit indirectly, as a vertex partitioning problem and further studied in [2,3,4,5,11]. A set S V is a restrained dominating set, if every vertex not in S is adjacent to a vertex in S and to a vertex in V À S. Every graph has a restrained dominating set, since S V is such a set.…”

Section: Introductionmentioning

confidence: 99%