Total restrained domination in trees

Hattingh, Johannes H.; Jonck, Elizabeth; Joubert, Ernst J.; Plummer, Andrew

doi:10.1016/j.disc.2006.09.014

Cited by 23 publications

(12 citation statements)

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“…In this paper, we continue the study of a variation of the domination theme, namely that of restrained domination -see [2,3,4,5,6,7,8,9,12,13].…”

Section: Introductionmentioning

confidence: 96%

“…Domke et al [4] provided a constructive characterization of trees achieving the lower bound given in Corollary 2. Hattingh and Plummer [9] gave a simpler characterization, independent of γ r -set consideration. In the sequel, we constructively characterize unicyclic graphs achieving the lower bound given in Corollary 3, utilizing constructive operations governed by status labeling.…”

Section: Introductionmentioning

confidence: 99%

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Restrained domination in unicyclic graphs

Hattingh

Joubert

Loizeaux

et al. 2009

Discuss. Math. Graph Theory

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Let G = (V, E) be a graph. A set S ⊆ V is a restrained dominating set if every vertex in V − S is adjacent to a vertex in S and to a vertex in V − S. The restrained domination number of G, denoted by γ r (G), is the minimum cardinality of a restrained dominating set of G. A unicyclic graph is a connected graph that contains precisely one cycle. We show that if U is a unicyclic graph of order n, then γ r (U) ≥ n 3 , and provide a characterization of graphs achieving this bound.

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“…In this paper, we continue the study of a variation of the domination theme, namely that of restrained domination -see [2,3,4,5,6,7,8,9,12,13].…”

Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

Restrained domination in unicyclic graphs

Hattingh

Joubert

Loizeaux

et al. 2009

Discuss. Math. Graph Theory

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“…The total domination number of G, denoted by γ t (G), is the minimum cardinality of a TDS of G, while the total restrained domination number of G, denoted by γ tr (G), is the minimum cardinality of a TRDS of G. Total domination in graphs is very well studied in graph theory (see, for example, the recent papers [8,[10][11][12]19]). Recent papers on total restrained domination in graphs can be found, for example, in [2,5,6,9,13,[15][16][17]21].…”

Section: Introductionmentioning

confidence: 99%

An Improved Upper Bound on the Total Restrained Domination Number in Cubic Graphs

Southey

Henning

2011

Graphs and Combinatorics

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In this paper, we continue the study of total restrained domination in graphs. A set S of vertices in a graph G = (V, E) is a total restrained dominating set of G if every vertex of G is adjacent to some vertex in S and every vertex of V \S is adjacent to a vertex in V \S. The minimum cardinality of a total restrained dominating set of G is the total restrained domination number γ tr (G) of G. Jiang et al. (Graphs Combin 25:341-350, 2009) showed that if G is a connected cubic graph of order n, then γ tr (G) ≤ 13n/19. In this paper we improve this upper bound to γ tr (G) ≤ (n + 4)/2. We provide two infinite families of connected cubic graphs G with γ tr (G) = n/2, showing that our new improved bound is essentially best possible.

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“…The concepts of restrained dominating and total restrained dominating sets are studied and investigated in [1], [2], [5], [6] and [7]. Other types of domination in graphs can be found in [3] and [4].…”

Section: Introductionmentioning

confidence: 99%