Nordhaus–Gaddum results for restrained domination and total restrained domination in graphs

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: 97%

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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“…In 1999, Domke et al [5] introduced and investigated the concept of restrained domination in graphs. In 2008, Hattingh et al [8] investigated the same concept and obtained a Nordhaus-Gaddum results for restrained domination and total restrained domination in graphs. Moreover, in 2015, Omega et al [10] introduced and characterized the restrained locating-dominating sets of some graphs and determined the restrained L-domination numbers of these graphs.…”

Section: Introductionmentioning

confidence: 99%

Resolving Restrained Domination in Graphs

Monsanto

Rara²

2021

Eur. J. Pure Appl. Math.

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Let G be a connected graph. Brigham et al. [3] defined a resolving dominating setas a set S of vertices of a connected graph G that is both resolving and dominating. A set S âŠ† V (G) is a resolving restrained dominating set of G if S is a resolving dominating set of G and S = V (G) or hV (G) \ Si has no isolated vertex. In this paper, we characterize the resolving restrained dominating sets in the join, corona and lexicographic product of graphs and determine the resolving restrained domination number of these graphs.

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Nordhaus–Gaddum results for restrained domination and total restrained domination in graphs

Cited by 31 publications

References 9 publications

Restrained domination in unicyclic graphs

Restrained domination in unicyclic graphs

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

Resolving Restrained Domination in Graphs

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