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Journal ArticleDOI

Restrained 2-Domination in Graphs

TL;DR: In this paper, the authors studied restrained 2domination in graphs and obtained some results, including the smallest cardinality of a 2dominating set of a set of vertices in a graph.
Abstract: Let G = (V, E) be a graph. A set SV(G) is a restrained 2dominating set if every vertex of ( ) \ V G S is adjacent to at least two vertices in S and every vertex of ( ) \ V G S is adjacent to a vertex in ( ) \ V G S . The restrained 2domination number of G, denoted by 2 r  (G), is the smallest cardinality of a restrained 2dominating set of G. In this paper we study restrained 2domination in graphs and obtain some results.

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Citations
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Journal ArticleDOI
TL;DR: In this article, the authors presented 2-dominating sets for complete grid graphs by using V-merge and Hmerge techniques recursively and then found 2domination and restrined 2dominance numbers of complete grid graph.
Abstract: In this paper we present 2-dominating sets for complete grid graphs $G_{k,n}$ by using V-merge and H-merge techniques recursively and then find 2-domination and restrined 2-domination numbers of complete grid graphs. We also verify the Vizing’s conjecture for both domination parameters.

7 citations


Cites methods from "Restrained 2-Domination in Graphs"

  • ...From [5], γr2(Pm)× γr2(Pn) = m× n and hence γr2(Pm × Pn) ≤ γr2(Pm)× γr2(Pn) for m,n ≥ 3....

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  • ...We have the following characterization of those graphs for which the 2-restrained domination number is equal to the 2-domination number [5]....

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  • ...The restrained 2-domination number of G denoted by γr2(G) is the minimum cardinality of a restrained 2-dominating set of G. Email address: j.jaganmohan@hotmail.com (J. Jagan Mohan) 352...

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  • ...The concept of restrained 2- domination was introduced by Kelkar and Mohan [5]....

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Journal ArticleDOI
TL;DR: In this article, the 2-dominating sets in the join, and the corona of graphs are characterized and their corresponding 2domination numbers are determined, and some sharp upper bounds for the restrained 2domdomination number of the lexicographic products of graphs were obtained.
Abstract: The concepts of 2-domination and restrained 2-domination are among the variations of the standard domination concept in a graph. In this paper, the 2-dominating sets in the join, and the corona of graphs are characterized and their corresponding 2-domination numbers are determined. The restrained 2-dominating sets in the lexicographic product of graphs are also characterized. Some sharp upper bounds for the restrained 2-domination number of the lexicographic products of graphs are obtained. Mathematics Subject Classification: 05C12

4 citations


Cites background from "Restrained 2-Domination in Graphs"

  • ...The concepts of restrained domination, 2-domination and restrained 2domination in graphs were studied in [1, 6], [3, 4] and [2, 5], respectively....

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Journal ArticleDOI
TL;DR: In this paper, the authors characterized the graphs obtained in the join of two graphs having 2,3,4 = ) ( 2 G r , and gave a formula for computing the restrained 2 -domination number.
Abstract: Let G be a graph. A subset S of ) (G V is a restrained 2-dominating set if every vertex of S G V \\ ) ( is adjacent to at least two vertices of S and every vertex of S G V \\ ) ( is adjacent to at least one vertex in S G V \\ ) ( . The restrained 2 -domination number of G , denoted by ) ( 2 G r  , is the smallest cardinality of a restrained 2 -dominating set G . In this paper, we characterize the graphs obtained in the join of two graphs having 2,3,4 = ) ( 2 G r  . We also characterize restrained 2 -dominating sets in the corona of two graphs and give a formula for computing the restrained 2 -domination number.

2 citations

Journal ArticleDOI
TL;DR: In this paper, the authors characterized the restrained independent 2-dominating sets of the join and corona of graphs and calculated their restrained independent two-domination numbers, where every vertex not in S is dominated at least twice and adjacent to at least one vertex in S, and every pair of vertices in S are not adjacent.
Abstract: A restrained independent 2-dominating set of a graph G is a set S of vertices of G such that every vertex not in S is dominated at least twice and adjacent to at least one vertex not in S, and every pair of vertices in S are not adjacent. In this paper, we characterized the restrained independent 2-dominating sets of the join and corona of graphs and calculate their restrained independent 2-domination numbers. Mathematics Subject Classification: 05C69
Proceedings ArticleDOI
01 Oct 2016
TL;DR: In this paper, the upper and lower bounds of the restrained-isolate dominating set of a simple graph G were characterized and some characterization of the domination in some binary operations were obtained.
Abstract: A dominating set S ⊆ V (G) of a simple graph G is called a restrained-isolate dominating set of G if the subgraph 〈S〉 induced by S has an isolated vertex and the subgraph 〈V (G) ∖ S〉 induced by V (G) ∖ S has no isolated vertex. A restrained-isolate dominating set of minimum cardinality is called restrained-isolate domination number of G, denoted by γ r0 (G). In this paper, we characterized the upper and lower bounds of the restrained-isolate domination. We also obtained some characterization of the restrained-isolate domination in some binary operations.

Cites background from "Restrained 2-Domination in Graphs"

  • ...Kelkar and Mohan in [4] introduced the concept of 2-domination and derived some characterization when the restrained domination number becomes equally the restrained 2-domination number....

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References
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Book
01 Jan 1969

16,023 citations

Book
01 Jan 1998
TL;DR: A survey of domination-related parameters topics on directed graphs graphs can be found in this article with respect to the domination number bondage, insensitivity, and reinforcement of graph dominating functions.
Abstract: LP-duality, complementarity and generality of graphical subset parameters dominating functions in graphs fractional domination and related parameters majority domination and its generalizations convexity of external domination-related functions of graphs combinatorial problems on chessboards - II domination in cartesian products - Vizing's conjecture algorithms complexity results domination parameters of a graph global domination distance domination in graphs domatic numbers of graphs and their variants - a survey domination-related parameters topics on domination in directed graphs graphs critical with respect to the domination number bondage, insensitivity and reinforcement.

1,289 citations

Journal ArticleDOI
TL;DR: A design methodology of practical solution algorithms for generally $\NP$-hard problems when restricted to partial k-trees (graphs with treewidth bounded by k) is presented, which accounts for dependency on the parameter k of the computational complexity of the resulting algorithms.
Abstract: In this paper, we consider a large class of vertex partitioning problems and apply to them the theory of algorithm design for problems restricted to partial k-trees. We carefully describe the details of algorithms and analyze their complexity in an attempt to make the algorithms feasible as solutions for practical applications. We give a precise characterization of vertex partitioning problems, which include domination, coloring and packing problems, and their variants. Several new graph parameters are introduced as generalizations of classical parameters. This characterization provides a basis for a taxonomy of a large class of problems, facilitating their common algorithmic treatment and allowing their uniform complexity classification. We present a design methodology of practical solution algorithms for generally $\NP$-hard problems when restricted to partial k-trees (graphs with treewidth bounded by k). This "practicality" accounts for dependency on the parameter k of the computational complexity of the resulting algorithms. By adapting the algorithm design methodology on partial k-trees to vertex partitioning problems, we obtain the first algorithms for these problems with reasonable time complexity as a function of treewidth. As an application of the methodology, we give the first polynomial-time algorithm on partial k-trees for computation of the Grundy number.

340 citations


"Restrained 2-Domination in Graphs" refers background in this paper

  • ...A vertex in a graph G dominates itself and its neighbors....

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Book
18 Sep 1985

212 citations


"Restrained 2-Domination in Graphs" refers background in this paper

  • ...Cyman and Raczek [6] introduced the concept of total restrained domination....

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Journal ArticleDOI
TL;DR: The study of a variation of standard domination, namely restrained domination, is initiated and it is shown that the decision problem for γ r ( G ) is NP-complete even for bipartite and chordal graphs.

140 citations